Tabla Integrales

17 TABLES of SPECIAL INDEFINITE INTEGRALS

Here we provide tables of special indefinite integrals. As stated in the remarks on page 67, here a, b, p, q, n are constants, restricted if indicated; e = 2.71828 . . . is the natural base of logarithms; ln u denotes the natural logarithm of u, where it is assumed that u > 0 (in general, to extend formulas to cases where u < 0 as well, replace ln u by ln |u|); all angles are in radians; and all constants of integration are omitted but implied. It is assumed in all cases that division by zero is excluded.

Our integrals are divided into types which involve the following algebraic expressions and functions:

(1) ax + b (13) ax bx c2 + + (25) eax

(2) ax b+ (14) x3 + a3 (26) ln x

(3) ax + b and px + q (15) x a4 4± (27) sinh ax

(4) ax b+ and px + q (16) x an n± (28) cosh ax

(5) ax b px q+ +and (17) sin ax (29) sinh ax and cosh ax

(6) x2 + a2 (18) cos ax (30) tanh ax

(7) x2 – a2, with x2 > a2 (19) sin ax and cos ax (31) coth ax

(8) a2 – x2, with x2 < a2 (20) tan ax (32) sech ax

(9) x a2 2+ (21) cot ax (33) csch ax

(10) x a2 2− (22) sec ax (34) inverse hyperbolic functions

(11) a x2 2− (23) csc ax

(12) ax2 + bx + c (24) inverse trigonometric functions

Some integrals contain the Bernouilli numbers Bn and the Euler numbers E

n defined in Chapter 23.

(1) Integrals Involving ax � b

17.1.1. dx

ax b aax b

+= +∫

1ln( )

17.1.2. x dx

ax bxa

ba

ax b+ = − +∫ 2 ln ( )

17.1.3.

x dxax b

ax ba

b ax ba

ba

ax b2 2

3 3

2

322

+= + − + + +( ) ( )

ln( )∫∫17.1.4.

dxx ax b b

xax b( )

ln+ = +⎛⎝⎜

⎞⎠⎟∫

1

17.1.5.

dxx ax b bx

ab

ax bx2 2

1( )

ln+ = − + +⎛⎝⎜

⎞⎠⎟∫

17.1.6.

dxax b a ax b( ) ( )+ = −

+∫ 2

1

17.1.7. x dx

ax bb

a ax b aax b

( ) ( )ln ( )+ = + + +∫ 2 2 2

1

17.1.8. x dx

ax bax b

ab

a ax bb

aax b

2

2 3

2

3 3

2( ) ( )

ln ( )+ = + − + − +∫

17.1.9. dxx ax b b ax b b

xax b( ) ( )

ln+ = + + +⎛⎝⎜

⎞⎠⎟∫ 2 2

1 1

71

72

17.1.10.

dxx ax b

ab ax b b x

ab

ax bx2 2 2 2 3

1 2( ) ( )

ln+ = −+ − + +⎛

⎝⎜⎞⎠⎠⎟∫

17.1.11.

dxax b ax b( ) ( )+ = −

+∫ 3 2

12

17.1.12.

x dxax b a ax b

ba ax b( ) ( ) ( )+ = −

+ + +∫ 3 2 2 2

12

17.1.13.

x dxax b

ba ax b

ba ax b a

a2

3 3

2

3 2 3

22

1( ) ( ) ( )

ln (+ = + − + + xx b+∫ )

17.1.14.

( )( )

( ). , . .ax b dx

ax bn a

nnn

+ = ++ = −

+1

11 17 1If see 11.∫

17.1.15.

x ax b dxax bn a

b ax bn

nn n

( )( )( )

( )(

+ = ++ − +

+∫+ +2

2

1

2 1)), ,

an2 1 2≠ − −

If n = –1, –2, see 17.1.2 and 17.1.7.

17.1.16.

x ax b dxax bn a

b ax bn

nn n

23

3

2

32

( )( )( )

( )(

+ = ++ − +

∫+ +

++ + ++

+

2 13

2 1

3)( )( )a

b ax bn a

n

If n = –1, –2, –3, see 17.1.3, 17.1.8, and 17.1.13.

17.1.17.

x ax b dx

x ax bm n

nbm n

x ax

m n

m nm

∫ + =

++ + + + + +

+

( )

( )(

1

1 1bb dx

x ax bm n a

mbm n a

x

n

m nm

)

( )( ) ( )

−

+−

∫+

+ + − + +

1

11

1 1(( )

( )( ) ( )

ax b dx

x ax bn b

m nn b

n

m n

+

− ++ + + +

+

∫+ +1 1

12

1xx ax b dxm n∫ +

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

+( ) 1

(2) Integrals Involving ax b�

17.2.1.

dx

ax b

ax ba+

= +∫2

17.2.2.

x dx

ax b

ax ba

ax b+

= − +∫2 2

3 2

( )

17.2.3.

x dx

ax b

a x abx ba

ax b2 2 2 2

3

2 3 4 815+

= − + +∫( )

17.2.4.dx

x ax b

b

ax b b

ax b b

b

ax b+=

+ −+ +

⎛

⎝⎜

⎞

⎠⎟

−+

−−

1

2 1

ln

tanbb

⎧

⎨

⎪⎪

⎩

⎪⎪

∫

17.2.5.dx

x ax b

ax bbx

ab

dx

x ax b2 2+= − + −

+∫∫ (see 17.2.12.)

17.2.6. ax b dxax b

a+ =

+∫

23

3( )

17.2.7. x ax b dxax b

aax b+ = − +∫

2 3 215 2

3( )( )

TABLES OF SPECIAL INDEFINITE INTEGRALS

17.2.8. x ax b dxa x abx b

aax b2

2 2 2

332 15 12 8

105+ = − + +∫

( )( )

17.2.9.ax b

xdx ax b b

dx

x ax b

+ = + ++∫∫ 2 (See 17.2.12.)

17.2.10.ax bx

dxax b

xa dx

x ax b

+ = − + ++∫ ∫2 2

(See 17.2.12.)

17.2.11.x

ax bdx

x ax bm a

mbm a

x

ax b

m m m

+= +

+ − + +∫−2

2 12

2 1

1

( ) ( )ddx∫

17.2.12.dx

x ax b

ax bm bx

m am b

dx

xm m m+= − +

− − −−−( )

( )( )12 32 21 −− +∫∫ 1 ax b

17.2.13. x ax b dxx

m aax b

mbm a

xmm

m+ = + + − +−2

2 32

2 33 2 1

( )( )

( )/ aax b dx+∫∫

17.2.14.ax bx

dxax b

m xa

mdx

x ax bm m m

+ = − +− + − +− −∫ ∫( ) ( )1 2 11 1

17.2.15.ax bx

dxax b

m bxm amm m

+ = − +− − −

−−( )

( )( )( )

/3 2

112 52 2 bb

ax bx

dxm

+−∫∫ 1

17.2.16. ( )( )

( )/

( )/

ax b dxax ba m

mm

+ = ++∫

+2

2 2

2

22

17.2.17. x ax b dxax ba m

b ax bmm

( )( )

( )(/

( )/

+ = ++ − +∫

+2

4 2

2

24

2 ))( )

( )/m

a m

+

+2 2

2 2

17.2.18. x ax b dxax ba m

b axmm

2 26 2

3

26

4( )

( )( )

(/( )/

+ = ++

− +∫+ bb

a mb ax b

a m

m m)( )

( )( )

( )/ ( )/+ +

++ +

+

4 2

3

2 2 2

342

2

17.2.19.( ) ( ) ( )/ / ( )/ax b

xdx

ax bm

bax b

xd

m m m+ = + + +∫ ∫

−2 2 2 22xx

17.2.20.( ) ( ) ( )/ ( )/ax b

xdx

ax bbx

mab

ax bm m m+ = − + + ++

∫2

2

2 2

2

//2

xdx∫

17.2.21.dx

x ax b m b ax b bdx

x axm m( ) ( ) ( ) (/ ( )/+ = − + + +−2 2 2

22

1bb m)( )/−∫∫ 2 2

(3) Integrals Involving ax � b and px � q

17.3.1.

dxax b px q bp aq

px qax b( )( )

ln+ + = −++

⎛⎝⎜

⎞⎠⎟∫

1

17.3.2.x dx

ax b px q bp aqba

ax bqp

px q( )( )

ln( ) ln( )+ +

=−

+ − +1 ⎧⎧⎨⎩

⎫⎬⎭∫

TABLES OF SPECIAL INDEFINITE INTEGRALS 73

74

17.3.3.

dxax b px q bp aq ax b

pbp aq

px qax( ) ( )

ln+ + = − + + −++2

1 1bb

⎛⎝⎜

⎞⎠⎟

⎧⎨⎩

⎫⎬⎭∫

17.3.4.

x dxax b px q bp aq

qbp aq

ax bpx q( ) ( )

ln+ + = − −++

⎛⎝⎜

⎞2

1⎠⎠⎟ − +

⎧⎨⎩

⎫⎬⎭∫

ba ax b( )

17.3.5.x dx

ax b px qb

bp aq a ax b bp aq

2

2

2

2

1( ) ( ) ( ) ( ) (+ +

=− +

+− ))

ln ( )( )

ln ( )2

2

2

2qp

px qb bp aq

aax b+ + − +

⎧⎨⎩

⎫⎬⎭

∫

17.3.6.dx

ax b px q n bp aq ax b pm n m( ) ( ) ( )( ) ( ) (+ + = −− − + −

11

11 xx q

a m ndx

ax b px q

n

m n

+{+ + − + + }

−

−

∫

∫

)

( )( ) ( )

1

12

17.3.7.ax bpx q

dxaxp

bp aqp

px q++ = + − +∫ 2 ln( )

17.3.8.( )( )

( )( )( )(

ax bpx q

dx

n bp aqax bpx

m

n

m

++

=

−− −

+ +11

1

+++ − − +

+⎧⎨⎩

⎫⎬⎭

−

− −∫qn m a

ax bpx q

dxn

m

n)( )

( )( )1 1

2

1(( )

( )( )

( )( )

n m pax bpx q

m bp aqax bm

n

m

− −++

+ − +−

−

1 1

1

(( )

( )( )

( )

px qdx

n pax bpx q

m

n

m

n

+⎧⎨⎩

⎫⎬⎭

−−

++

−

∫

−

11 1

aaax bpx q

dxm

n

( )( )

++

⎧⎨⎩

⎫⎬⎭

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

−

−∫

∫1

1

(4) Integrals Involving ax b� and px � q

17.4.1.

px q

ax bdx

apx aq bpa

ax b++

= + − +∫2 3 2

3 2

( )

17.4.2.

dx

px q ax b

bp aq p

p ax b bp aq

p ax b

( )

ln( )

( )

+ += −

+ − −

+∫

1

++ −

⎛

⎝⎜

⎞

⎠⎟

−+

−

⎧

⎨

⎪⎪

⎩

⎪ −

bp aq

aq bp p

p ax baq bp

2 1tan( )

⎪⎪

17.4.3.

ax bpx q

dx

ax bp

bp aq

p p

p ax b bp aq

p a++

=

+ +− + − −2

ln( )

( xx b bp aq

ax bp

aq bp

p p

p ax b

+ + −

⎛

⎝⎜

⎞

⎠⎟

+ −− +−

)

tan(2 2

1 ))aq bp−

⎧

⎨

⎪⎪

⎩

⎪⎪

∫

17.4.4. ( )( )

( ) (px q ax b dx

px q ax bn p

bp aqnn

+ + = + ++ + −+2

2 3 2

1

nn ppx q

ax b

n

+++∫∫ 3)

( )

17.4.5.

dx

px q ax b

ax bn aq bp px q

nn n( ) ( )( )( )

(

+ += +

− − + +−12

1

−−− − + +∫ ∫ −

32 1 1

)( )( ) ( )

an aq bp

dx

px q ax bn

17.4.6.

( ) ( )( )

( )px q

ax bdx

px q ax bn a

n aq bpn n++

= + ++ + −2

2 12

(( )( )

2 1

1

n apx q dx

ax b

n

++

+

−

∫∫

17.4.7.

ax bpx q

dxax b

n p px qa

n pn n

++ = − +

− + + −−( ) ( ) ( ) ( )1 2 11∫∫ ∫ + +−

dx

px q ax bn( ) 1


75

(5) Integrals Involving ax b� and px q+

17.5.1.dx

ax b px q

apa px q p ax b

ap

( )( )

ln ( ) ( )

t+ +=

+ + +( )

−

∫

2

2aan

( )( )

− − ++

⎧

⎨⎪⎪

⎩⎪⎪

1 p ax ba px q

17.5.2.x dx

ax b px q

ax b px qap

bp aqap

dx

( )( )

( )( )

(+ +=

+ +− +

∫ 2 aax b px q+ +∫ )( )

17.5.3. ( )( ) ( )( )(

ax b px q dxapx bp aq

apax b px q

b+ + = + + + + −24

pp aqap

dx

ax b px q

−+ +∫∫

)

( )( )

2

8

17.5.4.px qax b

dxax b px q

aaq bp

adx

ax b p

++ =

+ ++ −

+∫( )( )

( )(2 xx q+∫ )

17.5.5.dx

px q ax b px q

ax b

aq bp px q( ) ( )( ) ( )+ + += +

− +∫2

(6) Integrals Involving x2 � a2

17.6.1.dx

x a axa2 2

11+ = −∫ tan

17.6.2.x dx

x ax a

2 22 21

2+= +∫ ln ( )

17.6.3. x dxx a

x axa

2

2 21

+ = − −∫ tan

17.6.4. x dxx a

x ax a

3

2 2

2 22 2

2 2+ = − +∫ ln ( )

17.6.5. dxx x a a

xx a( )

ln2 2 2

2

2 2

12+

=+

⎛⎝⎜

⎞⎠⎟∫

17.6.6.dx

x x a a x axa2 2 2 2 3

11 1( )

tan+ = − − −∫

17.6.7.dx

x x a a x ax

x a3 2 2 2 2 4

2

2 2

12

12( )

ln+

= − −+

⎛⎝⎜

⎞⎠⎟∫

17.6.8.dx

x ax

a x a axa( ) ( )

tan2 2 2 2 2 2 31

21

2+ = + + −∫

17.6.9.x dx

x a x a( ) ( )2 2 2 2 2

12+ = −

+∫

17.6.10.x dx

x ax

x a axa

2

2 2 2 2 21

21

2( ) ( )tan+ = −

+ + −∫


76

17.6.11.x dx

x aa

x ax a

3

2 2 2

2

2 22 2

212( ) ( )

ln( )+

=+

+ +∫

17.6.12.dx

x x a a x a ax

x a( ) ( )ln2 2 2 2 2 2 4

2

2 2

12

12+ = + + +

⎛⎝⎜

⎞⎠⎟∫∫

17.6.13.dx

x x a a xx

a x a axa2 2 2 2 4 4 2 2 5

112

32( ) ( )

tan+ = − − + − −∫

17.6.14.dx

x x a a x a x a ax

x3 2 2 2 4 2 4 2 2 6

2

2

12

12

1( ) ( )

ln+ = − − + − + aa2

⎛⎝⎜

⎞⎠⎟∫

17.6.15.dx

x ax

n a x an

n an n( ) ( ) ( ) ( )2 2 2 2 2 1 22 12 3

2 2+ = − + + −−−∫∫ ∫ + −

dxx a n( )2 2 1

17.6.16.x dx

x a n x an n( ) ( )( )2 2 2 2 1

12 1+ = −

− + −∫

17.6.17.dx

x x a n a x a adx

x xn n( ) ( ) ( ) (2 2 2 2 2 1 2 2

12 1

1+ = − + + +∫ − aa n2 1) −∫

17.6.18.x dx

x ax dx

x aa

x dxx

m

n

m

n

m

( ) ( ) (2 2

2

2 2 12

2

2+ = + −−

−

−

∫∫ ++∫ a n2 )

17.6.19.

dxx x a a

dxx x a a

dxx xm n m n m( ) ( ) (2 2 2 2 2 1 2 2

1 1+ = + −∫ − − 22 2+∫∫ a n)

(7) Integrals Involving x2 � a2, x2 > a2

17.7.1.dx

x a ax ax a a

xa2 2

112

1− = −

+⎛⎝⎜

⎞⎠⎟ −∫ −ln cothor

17.7.2.x dx

x ax a2 2

2 212− = −∫ ln ( )

17.7.3.x dx

x ax

a x ax a

2

2 2 2− = + −+

⎛⎝⎜

⎞⎠⎟∫ ln

17.7.4.x dx

x ax a

x a3

2 2

2 22 2

2 2−= + −∫ ln( )

17.7.5.dx

x x a ax a

x( )ln2 2 2

2 2

2

12− = −⎛

⎝⎜⎞⎠⎟∫

17.7.6.dx

x x a a x ax ax a2 2 2 2 3

1 12( )

ln− = + −+

⎛⎝⎜

⎞⎠⎟∫

17.7.7.dx

x x a a x ax

x a3 2 2 2 2 4

2

2 2

12

12( )

ln− = − −⎛⎝⎜

⎞⎠⎟∫

17.7.8.dx

x ax

a x a ax ax a( ) ( )

ln2 2 2 2 2 2 321

4− = −− − −

+⎛⎝⎜

⎞⎠⎟∫


77

17.7.9. x dx

x a x a( ) ( )2 2 2 2 2

12− = −

−∫

17.7.10.x dx

x ax

x a ax ax a

2

2 2 2 2 221

4( ) ( )ln− = −

− + −+

⎛⎝⎜

⎞⎠⎟∫

17.7.11.x dx

x aa

x ax a

3

2 2 2

2

2 22 2

212( ) ( )

ln( )−

= −−

+ −∫

17.7.12.dx

x x a a x a ax

x a( ) ( )ln2 2 2 2 2 2 4

2

2 2

12

12− = −

− + −⎛⎝⎜

⎞⎠⎟⎟∫

17.7.13.dx

x x a a xx

a x a ax ax a2 2 2 2 4 4 2 2 5

12

34( ) ( )

ln− = − − − − −+

⎛⎛⎝⎜

⎞⎠⎟∫

17.7.14.dx

x x a a x a x a ax

x3 2 2 2 4 2 4 2 2 6

2

2

12

12

1( ) ( )

ln− = − − − + − aa2

⎛⎝⎜

⎞⎠⎟∫

17.7.15.dx

x ax

n a x an

n an n( ) ( ) ( ) ( )2 2 2 2 2 12 12 3

2 2− = −− − − −

−− 22 2 2 1∫ ∫ − −dx

x a n( )

17.7.16.x dx

x a n x an n( ) ( )( )2 2 2 2 1

12 1− = −

− − −∫

17.7.17. dxx x a n a x a a

dxx xn n( ) ( ) ( ) (2 2 2 2 2 1 2 2

12 1

1− = −

− − −−∫ −− −∫ a n2 1)

17.7.18.x dx

x ax dx

x aa

x dxx a

m

n

m

n

m

( ) ( ) (2 2

2

2 2 12

2

2− = − + −−

−

−

22 )n∫∫∫

17.7.19.dx

x x a adx

x x a adx

x x am n m n m( ) ( ) (2 2 2 2 2 2 2 2

1 1− = − − −− 22 1)n−∫∫∫

(8) Integrals Involving x2 � a2, x2 < a2

17.8.1.dx

a x aa xa x a

xa2 2

112

1− = +

−⎛⎝⎜

⎞⎠⎟∫ −ln tanhor

17.8.2.x dx

a xa x2 2

2 212− = − −∫ ln ( )

17.8.3.x dx

a xx

a a xa x

2

2 2 2− = − + +−

⎛⎝⎜

⎞⎠⎟∫ ln

17.8.4.x dx

a xx a

a x3

2 2

2 22 2

2 2−= − − −∫ ln( )

17.8.5.dx

x a x ax

a x( )ln2 2 2

2

2 2

12− = −

⎛⎝⎜

⎞⎠⎟∫

17.8.6.dx

x a x a x aa xa x2 2 2 2 3

1 12( )

ln− = − + +−

⎛⎝⎜

⎞⎠⎟∫


78

17.8.7. dx

x a x a x ax

a x3 2 2 2 2 4

2

2 2

12

12( )

ln− = − + −⎛⎝⎜

⎞⎠⎟∫

17.8.8. dx

a xx

a a x aa xa x( ) ( )

ln2 2 2 2 2 2 321

4− = − + +−

⎛⎝⎜

⎞⎠⎟∫

17.8.9. x dx

a x a x( ) ( )2 2 2 2 2

12− = −∫

17.8.10.x dx

a xx

a x aa xa x

2

2 2 2 2 221

4( ) ( )ln− = − − +

−⎛⎝⎜

⎞⎠⎟∫

17.8.11.x dx

a xa

a xa x

3

2 2 2

2

2 22 2

212( ) ( )

ln( )−

=−

+ −∫

17.8.12.dx

x a x a a x ax

a x( ) ( )ln2 2 2 2 2 2 4

2

2 2

12

12− = − + −

⎛⎝⎜

⎞⎠⎟∫∫

17.8.13.dx

x a x a xx

a a x aa xa x2 2 2 2 4 4 2 2 5

12

34( ) ( )

ln− = − + − + +−

⎛⎛⎝⎜

⎞⎠⎟∫

17.8.14.dx

x a x a x a a x ax

a3 2 2 2 4 2 4 2 2 6

2

2

12

12

1( ) ( )

ln− = − + − + − xx2

⎛⎝⎜

⎞⎠⎟∫

17.8.15.dx

a xx

n a a xn

n an n( ) ( ) ( ) ( )2 2 2 2 2 1 22 12 3

2 2− = − − + −−−∫∫ ∫ − −

dxa x n( )2 2 1

17.8.16.x dx

a x n a xn n( ) ( )( )2 2 2 2 1

12 1− = − − −∫

17.8.17.dx

x a x n a a x adx

x a xn n( ) ( ) ( ) (2 2 2 2 2 1 2 2

12 1

1− = − − + −− 22 1)n−∫∫

17.8.18.x dx

a xa

x dxa x

x dxa x

m

n

m

n

m

( ) ( ) (2 22

2

2 2

2

2− = − − −∫ ∫− −

22 1)n−∫

17.8.19.dx

x a x adx

x a x adx

x am n m n m( ) ( ) (2 2 2 2 2 1 2 2

1 1− = − +− −∫ 22 2−∫∫ x n)

(9) Integrals Involving x a2 2+

17.9.1. dx

x ax x a

xa2 2

2 2 1

+= + +∫ −ln( ) sinhor

17.9.2.x dx

x ax a

2 2

2 2

+= +∫

17.9.3.x dx

x a

x x a ax x a

2

2 2

2 2 22 2

2 2+= + − + +∫ ln( )

17.9.4.x dx

x a

x aa x a

3

2 2

2 2 3 22 2 2

3+= + − +∫

( ) /


79

17.9.5. dx

x x a aa x a

x2 2

2 21

+= − + +⎛

⎝⎜⎞

⎠⎟∫ ln

17.9.6. dx

x x a

x aa x2 2 2

2 2

2+= − +

∫

17.9.7. dx

x x a

x aa x a

a x ax3 2 2

2 2

2 2 3

2 2

21

2+= − + + + +⎛

⎝⎜⎞

⎠⎟∫ ln

17.9.8. x a dxx x a a

x x a2 22 2 2

2 2

2 2+ = + + + +∫ ln( )

17.9.9. x x a dxx a2 2

2 2 3 2

3+ = +∫

( ) /

17.9.10. x x a dxx x a a x x a a

x x2 2 22 2 3 2 2 2 2 4

2

4 8 8+ = + − + − +( )

ln(/

++∫ a2 )

17.9.11. x x a dxx a a x a3 2 2

2 2 5 2 2 2 2 3 2

5 3+ = + − +∫

( ) ( )/ /

17.9.12. x ax

dx x a aa x a

x

2 22 2

2 2+ = + − + +⎛

⎝⎜⎞

⎠⎟∫ ln

17.9.13.x a

xdx

x ax

x x a2 2

2

2 22 2+ = − + + + +∫ ln( )

17.9.14.x a

xdx

x ax a

a x ax

2 2

3

2 2

2

2 2

21

2+ = − + − + +⎛

⎝⎜⎞

⎠⎟∫ ln

17.9.15.dx

x ax

a x a( ) /2 2 3 2 2 2 2+ =+∫

17.9.16.x dx

x a x a( ) /2 2 3 2 2 2

1+ = −

+∫

17.9.17.x dx

x ax

x ax x a

2

2 2 3 2 2 2

2 2

( )ln ( )/+ = −

++ + +∫

17.9.18.x dx

x ax a

a

x a

3

2 2 3 22 2

2

2 2( ) /+ = + ++∫

17.9.19.dx

x x a a x a aa x a

x( )ln/2 2 3 2 2 2 2 3

2 21 1+ =

+− + +⎛

⎝⎜⎞

⎠⎟∫

17.9.20.dx

x x ax aa x

x

a x a2 2 2 3 2

2 2

4 4 2 2( ) /+ = − + −+∫

17.9.21.dx

x x a a x x a a x a a3 2 2 3 2 2 2 2 2 4 2 2 5

1

2

3

2

32( )

l/+ = −+

−+

+ nna x a

x+ +⎛

⎝⎜⎞

⎠⎟∫2 2

17.9.22. ( )( )

l//

x a dxx x a a x x a

a2 2 3 22 2 3 2 2 2 2

4

43

838

+ = + + + +∫ nn( )x x a+ +2 2

17.9.23. x x a dxx a

( )( )/

/2 2 3 2

2 2 5 2

5+ = +∫


80

17.9.24. x x a dxx x a a x x a2 2 2 3 2

2 2 5 2 2 2 2 3 2

6 2( )

( ) ( )// /

+ = + − +∫ 44 16 16

4 2 2 62 2− + − + +a x x a a

x x aln ( )

17.9.25. x x a dxx a a x a3 2 2 3 2

2 2 7 2 2 2 2 5 2

7 5( )

( ) ( )// /

+ = + − +∫

17.9.26.( ) ( )

ln/ /x a

xdx

x aa x a a

a x2 2 3 2 2 2 3 22 2 2 3

2

3+ = + + + − + + aa

x

2⎛

⎝⎜⎞

⎠⎟∫

17.9.27.( ) ( )

ln/ /x a

xdx

x ax

x x aa

2 2 3 2

2

2 2 3 2 2 223

232

+ = − + + + + (( )x x a+ +∫ 2 2

17.9.28.( ) ( )

ln/ /x a

xdx

x ax

x a a2 2 3 2

3

2 2 3 2

22 2

232

32

+ = − + + + − aa x ax

+ +⎛

⎝⎜⎞

⎠⎟∫2 2

(10) Integrals Involving x a2 2−

17.10.1.dx

x ax x a

x dx

x ax a

2 2

2 2

2 2

2 2

−= + −

−= −∫ ∫ln( ),

17.10.2.x dx

x a

x x a ax x a

2

2 2

2 2 22 2

2 2−= − + + −∫ ln ( )

17.10.3.x dx

x a

x aa x a

3

2 2

2 2 3 22 2 2

3−= − + −∫

( ) /

17.10.4.dx

x x a axa2 2

11

−= −∫ sec

17.10.5.dx

x x a

x aa x2 2 2

2 2

2−= −

∫

17.10.6.dx

x x a

x aa x a

xa3 2 2

2 2

2 2 31

21

2−= − + −∫ sec

17.10.7. x a dxx x a a

x x a2 22 2 2

2 2

2 2− = − − + −∫ ln ( )

17.10.8. x x a dxx a2 2

2 2 3 2

3− = −∫

( ) /

17.10.9. x x a dxx x a a x x a a

x x2 2 22 2 3 2 2 2 2 4

2

4 8 8− = − + − − +( )

ln (/

−−∫ a2 )

17.10.10. x x a dxx a a x a3 2 2

2 2 5 2 2 2 2 3 2

5 3− = − + −∫

( ) ( )/ /

17.10.11.x a

xdx x a a

xa

2 22 2 1− = − − −∫ sec

17.10.12.x a

xdx

x ax

x x a2 2

2

2 22 2− = − − + + −∫ ln ( )


81

17.10.13.x a

xdx

x ax a

xa

2 2

3

2 2

21

21

2− = − − + −∫ sec

17.10.14.dx

x ax

a x a( ) /2 2 3 2 2 2 2− = −−∫

17.10.15.x dx

x a x a( ) /2 2 3 2 2 2

1− = −

−∫

17.10.16.x dx

x ax

x ax x a

2

2 2 3 2 2 2

2 2

( )ln( )

/−= −

−+ + −∫

17.10.17. x dxx a

x aa

x a

3

2 2 3 22 2

2

2 2( ) /− = − −−∫

17.10.18.dx

x x a a x a axa( )

sec/2 2 3 2 2 2 2 3

11 1−

= −−

− −∫

17.10.19.dx

x x ax aa x

x

a x a2 2 2 3 2

2 2

4 4 2 2( ) /− = − − −−∫

17.10.20.dx

x x a a x x a a x a a3 2 2 3 2 2 2 2 2 4 2 2 5

1

2

3

2

32( )

se/−

=−

−−

− cc−∫ 1 xa

17.10.21. ( )( )

l//

x a dxx x a a x x a

a2 2 3 22 2 3 2 2 2 2

4

43

838

− = − − − +∫ nn( )x x a+ −2 2

17.10.22. x x a dxx a

( )( )/

/2 2 3 2

2 2 5 2

5− = −∫

17.10.23. x x a dxx x a a x x a2 2 2 3 2

2 2 5 2 2 2 2 3 2

6 2( )

( ) ( )// /

− = − + −∫ 44 16 16

4 2 2 62 2− − + + −a x x a a

x x aln ( )

17.10.24. x x a dxx a a x a3 2 2 3 2

2 2 7 2 2 2 2 5 2

7 5( )

( ) ( )// /

− = − + −∫

17.10.25.( ) ( )

sec/ /x a

xdx

x aa x a a

x2 2 3 2 2 2 3 22 2 2 3 1

3− = − − − +∫ −

aa

17.10.26.( ) ( )

l/ /x a

xdx

x ax

x x aa

2 2 3 2

2

2 2 3 2 2 223

232

− = − − + − −∫ nn ( )x x a+ −2 2

17.10.27.( ) ( )

se/ /x a

xdx

x ax

x aa

2 2 3 2

3

2 2 3 2

2

2 2

23

232

− = − − + − − cc−∫ 1 xa

(11) Integrals Involving a x2 2−

17.11.1.dx

a x

xa2 2

1

−= −∫ sin

17.11.2.x dx

a xa x

2 2

2 2

−= − −∫


82

17.11.3. x dx

a x

x a x a xa

2

2 2

2 2 21

2 2−= − − +∫ −sin

17.11.4. x dx

a x

a xa a x

3

2 2

2 2 3 22 2 2

3−= − − −∫

( ) /

17.11.5. dx

x a x aa a x

x2 2

2 21

−= − + −⎛

⎝⎜⎞

⎠⎟∫ ln

17.11.6. dx

x a x

a xa x2 2 2

2 2

2−= − −

∫

17.11.7. dx

x a x

a xa x a

a a xx3 2 2

2 2

2 2 3

2 2

21

2−= − − − + −⎛

⎝⎜⎞

⎠⎟∫ ln

17.11.8. a x dxx a x a x

a2 2

2 2 21

2 2− = − +∫ −sin

17.11.9. x a x dxa x2 2

2 2 3 2

3− = − −∫

( ) /

17.11.10. x a x dxx a x a x a x a2 2 2

2 2 3 2 2 2 2 41

4 8 8− = − − + − +∫ −( )

sin/ xx

a

17.11.11. x a x dxa x a a x3 2 2

2 2 5 2 2 2 2 3 2

5 3− = − − −∫

( ) ( )/ /

17.11.12.a x

xdx a x a

a a xx

2 22 2

2 2− = − − + −⎛

⎝⎜⎞

⎠⎟∫ ln

17.11.13.a x

xdx

a xx

xa

2 2

2

2 21− = − − − −∫ sin

17.11.14.a x

xdx

a xx a

a a xx

2 2

3

2 2

2

2 2

21

2− = − − + + −⎛

⎝⎜⎞

⎠⎟∫ ln

17.11.15.dx

a xx

a a x( ) /2 2 3 2 2 2 2− =−∫

17.11.16.x dx

a x a x( ) /2 2 3 2 2 2

1− =

−∫

17.11.17.x dx

a xx

a x

xa

2

2 2 3 2 2 2

1

( )sin/− =

−−∫ −

17.11.18.x dx

a xa x

a

a x

3

2 2 3 22 2

2

2 2( ) /− = − +−∫

17.11.19.dx

x a x a a x aa a x

x( )ln/2 2 3 2 2 2 2 3

2 21 1− =

−− + −⎛

⎝⎜⎞

⎠⎟∫

17.11.20.dx

x a xa xa x

x

a a x2 2 2 3 2

2 2

4 4 2 2( ) /− = − − +−∫

17.11.21.dx

x a x a x a x a a x a3 2 2 3 2 2 2 2 2 4 2 2 5

1

2

3

2

32( ) /− = −

−+

−−∫ lln

a a xx

+ −⎛

⎝⎜⎞

⎠⎟2 2


83

17.11.22. ( )( )

s//

a x dxx a x a x a x

a2 2 3 22 2 3 2 2 2 2

4

43

838

− = − + − +∫ iin−1 xa

17.11.23. x a x dxa x

( )( )/

/2 2 3 2

2 2 5 2

5− = − −∫

17.11.24. x a x dxx a x a x a x2 2 2 3 2

2 2 5 2 2 2 2 3 2

6( )

( ) ( )// /

− = − − + −∫ 224 16 16

4 2 2 61+ − + −a x a x a x

asin

17.11.25. x a x dxa x a a x3 2 2 3 2

2 2 7 2 2 2 2 5 2

7 5( )

( ) ( )// /

− = − − −∫

17.11.26.( ) ( )

ln/ /a x

xdx

a xa a x a

a a2 2 3 2 2 2 3 22 2 2 3

2

3− = − + − − +

∫−−⎛

⎝⎜

⎞

⎠⎟

xx

2

17.11.27.( ) ( )

s/ /a x

xdx

a xx

x a xa

2 2 3 2

2

2 2 3 2 2 223

232

− = − − − − −∫ iin−1 xa

17.11.28.( ) ( )

l/ /a x

xdx

a xx

a xa

2 2 3 2

3

2 2 3 2

2

2 2

23

232

− = − − − − +∫ nna a x

x+ −⎛

⎝⎜⎞

⎠⎟2 2

(12) Integrals Involving ax2 � bx � c

17.12.1.dx

ax bx c

ac b

ax b

ac b

b ac

2

2

1

2

2

2

4

2

4

1

4

+ +=

−+−

−

−tan

ln22 4

2 4

2

2

ax b b ac

ax b b ac

+ − −+ + −

⎛

⎝⎜

⎞

⎠⎟

⎧

⎨⎪⎪

⎩⎪⎪

∫

If b ac ax bx c a x b a2 2 24 2= + + = +, ( / ) and the results 17.1.6 to 17.1.10 and 17.1.14 to 17.1.17 can be used. If b = 0 use results on page 75. If a or c = 0 use results on pages 71–72.

17.12.2.x dx

ax bx c aax bx c

ba

dxax bx c2

22

12 2+ + = + + − + +∫∫ ln ( )

17.12.3.x dx

ax bx cxa

ba

ax bx cb ac

a

2

2 22

2

222

2+ + = − + + + −∫ ln ( )

ddxax bx c2 + +∫

17.12.4.x dx

ax bx cx

m aca

x dxax bx c

ba

m m m

2

1 2

21+ + = − − + + −− −

( )xx dx

ax bx c

m−

+ +∫∫∫1

2

17.12.5.dx

x ax bx c cx

ax bx cbc

dxa( )

ln2

2

2

12 2+ + = + +

⎛⎝⎜

⎞⎠⎟

−∫ xx bx c2 + +∫

17.12.6.dx

x ax bx cbc

ax bx cx cx2 2 2

2

221

( )ln+ + = + +⎛

⎝⎜⎞⎠⎟

− +∫bb ac

cdx

ax bx c

2

2 2

22−

+ +∫

17.12.7.dx

x ax bx c n cxbc

dxx ax bxn n n( ) ( ) (2 1 1 2

11+ + = − − − +− − ++ − + +−∫∫∫ c

ac

dxx ax bx cn) ( )2 2

17.12.8.dx

ax bx cax b

ac b ax bx ca

ac( ) ( )( )2 2 2 2

24

24+ + = +

− + + + −− + +∫∫ bdx

ax bx c2 2

17.12.9.x dx

ax bx cbx c

ac b ax bx cb

a( ) ( )( )2 2 2 2

24 4+ + = − +

− + + −cc b

dxax bx c− + +∫ ∫2 2


84

17.12.10.x dx

ax bx cb ac x bc

a ac b ax b

2

2 2

2

2 2

24( )

( )( )(+ + = − +

− + xx cc

ac bdx

ax bx c+ + − + +∫ ∫)2

4 2 2

17.12.11.x dx

ax bx cx

n m a ax bx c

m

n

m

n( ) ( ) ( )2

1

2 12 1+ + = − − − + +−

− ++ −− − + +

− −

−

∫∫( )

( ) ( )

( )

m cn m a

x dxax bx c

n m b

m

n

12 1

2

2

(( ) ( )2 1

1

2n m ax dx

ax bx c

m

n− − + +−

∫

17.12.12.x dx

ax bx c ax dx

ax bx ccn

n

n

n

2 1

2

2 3

2 1

1− −

−+ + = + + −( ) ( ) aa

x dxax bx c

ba

x dxax bx c

n

n

n

n∫ ∫− −

+ + − + +2 3

2

2 2

2( ) ( )∫∫∫

17.12.13.dx

x ax bx c c ax bx cbc

dxax bx( ) ( ) (2 2 2 2

12 2+ + = + + − + +∫ cc c

dxx ax bx c) ( )2 2

1+ + +∫∫

17.12.14.dx

x ax bx c cx ax bx cac

dxax b2 2 2 2 2

1 3( ) ( ) (+ + = − + + − +∫ xx c

bc

dxx ax bx c+ − + +∫∫ ) ( )2 2 2

2

17.12.15.dx

x ax bx c m cx ax bx cm n m n( ) ( ) ( )(

2 1 2 1

11+ + = − − + + −− −

mm n am c

dxx ax bx c

m n b

m n

+ −− + +

− + −

∫ ∫ −2 3

1

2

2 2

)( ) ( )

( )(( ) ( )m c

dxx ax bx cm n− + +−∫1 1 2

(13) Integrals Involving ax bx c2 � �

In the following results if b ac ax bx c a x b a2 24 2= + + = +, ( / ) and the results 17.1 can be used. If b = 0 use the results 17.9. If a = 0 or c = 0 use the results 17.2 and 17.5.

17.13.1.dx

ax bx c

aa ax bx c ax b

a

2

2

1

12 2

1 2+ +=

+ + + +

−−

−

ln ( )

sinaax b

b ac a

ax b

ac b

+−

⎛⎝⎜

⎞⎠⎟

+−

⎛⎝⎜

⎞⎠

−2

1

24

1 2

4or sinh ⎟⎟

⎧

⎨⎪⎪

⎩⎪⎪

∫

17.13.2.x dx

ax bx c

ax bx ca

ba

dx

ax bx c2

2

22+ += + + −

+ +∫ ∫

17.13.3.x dx

ax bx c

ax ba

ax bx cb ac

ad2

2 22

2

2

2 34

3 48+ +

= − + + + −∫

xx

ax bx c2 + +∫

17.13.4.dx

x ax bx c

c

c ax bx c bx cx

2

21 2 2

1+ +=

− + + + +⎛

⎝⎜⎞

⎠⎟

−

∫ln

cc

bx c

x b ac c

bx csin

| |sinh− −+

−⎛⎝⎜

⎞⎠⎟

− +1

2

12

4

1 2or

|| |x ac b4 2−⎛⎝⎜

⎞⎠⎟

⎧

⎨⎪⎪

⎩⎪⎪

17.13.5.dx

x ax bx c

ax bx ccx

bc

dx

x ax bx c2 2

2

22+ += − + + −

+ +∫ ∫

17.13.6. ax bx c dxax b ax bx c

aac b

adx

ax2

2 2

2

24

48

+ + = + + + + −∫( )

++ +∫ bx c


85

17.13.7. x ax bx c dxax bx c

ab ax b

aax2

2 3 2

22

328

+ + = + + − + +( ) ( )/

bbx c

b ac ba

dx

ax bx c

+

− −+ +

∫

∫( )4

16

2

2 2

17.13.8. x ax bx c dxax b

aax bx c

b a2 22

2 3 226 5

245 4+ + = − + + + −

( ) / cca

ax bx c dx16 2

2∫ ∫ + +

17.13.9.ax bx c

xdx ax bx c

b dx

ax bx cc

dx

x ax

22

2 22+ + = + + +

+ ++∫ ∫ ++ +∫

bx c

17.13.10.ax bx c

xdx

ax bx cx

adx

ax bx c

b dx

x a

2

2

2

2 2+ + = − + + +

+ ++∫

xx bx c2 + +∫∫

17.13.11.dx

ax bx cax b

ac b ax bx c( )( )

( )/2 3 2 2 2

2 2

4+ + = +− + +∫

17.13.12.x dx

ax bx cbx c

b ac ax bx c( )( )

( )/2 3 2 2 2

2 2

4+ + = +− + +∫

17.13.13.x dx

ax bx cb ac x bc

a ac b ax

2

2 3 2

2

2

2 4 2

4( )( )

( )/+ + = − +

− 22 2

1

+ ++

+ +∫∫ bx c adx

ax bx c

17.13.14.dx

x ax bx c c ax bx c cdx

x ax bx c

b( ) /2 3 2 2 2

1 1+ + =

+ ++

+ +−

22 2 3 2cdx

ax bx c( ) /+ +∫∫∫

17.13.15.dx

x ax bx cax bx c

c x ax bx c

b2 2 3 2

2

2 2

22( ) /+ + = − + +

+ ++∫

−−+ +

−+ +

∫

∫

22

32

2 2 3 2

2 2

acc

dxax bx c

bc

dx

x ax bx c

( ) /

17.13.16. ( )( )( )/

/

ax bx c dxax b ax bx c

an

n2 1 2

2 1 224

+ + = + + +++

∫ (( )( )( )

( )( ) /

nn ac b

a nax bx c n

++ + −

++ + −

12 1 4

8 1

22 1 2 ddx∫

17.13.17. x ax bx c dxax bx c

a nn

n

( )( )

( )/

/2 1 2

2 3 2

2 3+ + = + +

+−+

+

∫bba

ax bx c dxn

22 1 2( ) /+ + +∫

17.13.18.dx

ax bx cax b

n ac b axn( )( )

( )( )(/2 1 2 2

2 22 1 4+ + = +

− −+ 22 1 2

2 2

8 12 1 4

+ +

+ −− − +

−∫ bx c

a nn ac b

dxax

n)

( )( )( ) (

/

bbx c n+ −∫ ) /1 2

17.13.19.dx

x ax bx c n c ax bx cn n( ) ( ) ( )/ /2 1 2 2 1 2

12 1+ + = − + ++ −∫

++ + + − + +− +∫1

22 1 2 2 1cdx

x ax bx cbc

dxax bx cn n( ) ( )/ /22∫


86


Note that for formulas involving x3 – a3 replace a with –a.

17.14.1. dx

x a ax a

x ax a a3 3 2

2

2 2 2

16

1

3+= +

− +⎛⎝⎜

⎞⎠⎟ + −ln

( )tan 11 2

3

x a

a

−∫

17.14.2.x dx

x a ax ax a

x a a3 3

2 2

211

61

3+= − +

+⎛⎝⎜

⎞⎠⎟

+ −ln( )

tan22

3

x a

a

−∫

17.14.3.x dx

x ax a

2

3 33 31

3+ = +∫ ln ( )

17.14.4. dx

x x a ax

x a( )ln3 3 3

3

3 3

13+ = +

⎛⎝⎜

⎞⎠⎟∫

17.14.5. dx

x x a a x ax ax a

x a2 3 3 3 4

2 2

2

1 16( )

ln( )+

= − − − ++

⎛⎝⎜

⎞⎠⎟⎟

− −∫ −1

3

2

34

1

a

x a

atan

17.14.6. dx

x ax

a x a ax a

x ax a( ) ( )ln

( )3 3 2 3 3 3 5

2

2 231

9+=

++ +

− +⎛⎛⎝⎜

⎞⎠⎟ + −−∫

2

3 3

2

35

1

a

x a

atan

17.14.7. x dx

x ax

a x a ax ax a

x( ) ( )ln

(3 3 2

2

3 3 3 4

2 2

31

18+=

++ − +

+ aa a

x a

a)tan

2 4

11

3 3

2

3

⎛⎝⎜

⎞⎠⎟

+ −−∫

17.14.8. x dx

x a x a

2

3 3 2 3 3

13( ) ( )+ = − +∫

17.14.9. dx

x x a a x a ax

x a( ) ( )ln3 3 2 3 3 3 6

3

3 3

13

13+ = + + +

⎛⎝⎜

⎞⎠⎟∫∫

17.14.10.dx

x x a a xx

a x a ax dx

x2 3 3 2 6

2

6 3 3 6 3

13

43( ) ( )+

= − −+

−+∫ aa3

(See 17.14.2.)∫

17.14.11.x dx

x axm

ax dxx a

m m m

3 3

23

3

3 32+=

−−

+

− −

∫ ∫

17.14.12.dx

x x a a n x adx

x x an n n( ) ( ) ( )3 3 3 1 3 3 3 3

11

1+ = −

− − +− −∫ ∫∫


17.15.1.dx

x a a

x ax a

x ax a4 4 3

2 2

2 2

1

4 2

2

2

1

2+ = + +− +

⎛⎝⎜

⎞⎠⎟

−∫ lnaa

xa

xa3

1 1

21

21

2tan tan− −−

⎛⎝⎜

⎞⎠⎟

− +⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥⎥

17.15.2.x dx

x a axa4 4 2

12

2

12+ = −∫ tan

17.15.3.x dx

x a a

x ax a

x ax a

2

4 4

2 2

2 2

1

4 2

2

2

1+ = − +

+ +⎛⎝⎜

⎞⎠⎟

−∫ ln22 2

12

121 1

a

xa

xa

tan tan− −−⎛⎝⎜

⎞⎠⎟

− +⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥⎥


87

17.15.4. x dx

x ax a

3

4 44 41

4+ = +∫ ln ( )

17.15.5. dx

x x a ax

x a( )ln4 4 4

4

4 4

14+ = +

⎛⎝⎜

⎞⎠⎟∫

17.15.6.dx

x x a a x a

x ax a

x ax a2 4 4 4 5

2 2

2 2

1 1

4 2

2

2( )ln+ = − − − +

+ +⎛⎛⎝⎜

⎞⎠⎟

+ −⎛⎝⎜

⎞⎠⎟

− +⎛⎝

∫

− −1

2 21

21

25

1 1

a

xa

xa

tan tan ⎜⎜⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

17.15.7. dx

x x a a x axa3 4 4 4 2 6

12

2

12

12( )

tan+ = − − −∫

17.15.8. dx

x a ax ax a a

xa4 4 3 3

114

12− = −

+⎛⎝⎜

⎞⎠⎟ −∫ −ln tan

17.15.9. x dx

x a ax ax a4 4 2

2 2

2 2

14− = −

+⎛⎝⎜

⎞⎠⎟∫ ln

17.15.10.x dx

x a ax ax a a

xa

2

4 411

41

2− = −+

⎛⎝⎜

⎞⎠⎟ +∫ −ln tan

17.15.11.x dx

x ax a

3

4 44 41

4−= −∫ ln( )

17.15.12.dx

x x a ax a

x( )ln4 4 4

4 4

4

14− = −⎛

⎝⎜⎞⎠⎟∫

17.15.13.dx

x x a a x ax ax a a3 4 4 4 5 5

1 14

12( )

ln tan− = + −+

⎛⎝⎜

⎞⎠⎟ + −−∫ 1 x

a

17.15.14.dx

x x a a x ax ax a3 4 4 4 2 6

2 2

2 2

12

14( )

ln− = + −+

⎛⎝⎜

⎞⎠⎟∫

(16) Integrals Involving xn � an

17.16.1.dx

x x a nax

x an n n

n

n n( )ln

+=

+⎛⎝⎜

⎞⎠⎟∫

1

17.16.2.x dxx a n

x an

n nn n

−

+ = +∫1 1

ln ( )

17.16.3.x dx

x ax dx

x aa

x dxx

m

n n r

m n

n n rn

m n

n( ) ( ) (+ = + − +−

−

−

∫ 1 aan r)∫∫

17.16.4.dx

x x a adx

x x a adx

x xm n n r n m n n r n m n( ) ( ) (+ = + −− −∫1 1

1 nn n ra+∫∫ )

17.16.5.dx

x x a n a

x a a

x a an n n

n n n

n n n+= + −

+ +⎛

⎝⎜⎞

⎠⎟∫1

ln

17.16.6.dx

x x a nax a

xn n n

n n

n( )ln− = −⎛

⎝⎜⎞⎠⎟∫

1


88

17.16.7. x dxx a n

x an

n nn n

−

−= −∫

1 1ln( )

17.16.8.x dx

x aa

x dxx a

x dxx a

m

n n rn

m n

n n r

m n

n n( ) ( ) ( )− = − + −− −

rr−∫∫∫ 1

17.16.9. dx

x x a adx

x x a adx

x x am n n r n m n n n r n m n( ) ( ) (−=

−−

−−

1 1nn r) −∫∫∫

17.16.10.dx

x x a n a

axn n n

n

n−= −∫

2 1cos

17.16.11.x dx

x a mak p

mx ap

m m m p

−

−−

+= − +∫

1

2 2 211 2 1

2sin

( )tan

π ccos[( ) / ]sin[( ) / ]

2 1 22 1 21

k ma k mk

m −−

⎛⎝⎜

⎞⎠⎟=

∑ ππ

−− − + −−

=∑1

22 1

22

22

1

2

mak p

mx ax

km p

k

m

cos( )

cos(π

ln11

22)π

ma+

⎛⎝⎜

⎞⎠⎟

where 0 < p � 2m.

17.16.12.x dx

x a makpm

x axkm

p

m m m p

−

−−= −

1

2 2 221

22cos ln cos

π π ++⎛⎝⎜

⎞⎠⎟

− −

=

−

−−

∑∫ a

makpm

x a

k

m

m p

2

1

1

211

sin tancoπ ss( / )

sin( / )

{ln

k ma k m

ma

k

m

m p

ππ

⎛⎝⎜

⎞⎠⎟

+

=

−

−

∑1

1

2

12

(( ) ( ) ln( )}x a x ap− + − +1

where 0 < p � 2m.

17.16.13.x dx

x a m a

p

m m

p

m p

−

+ +

−

− ++= −

+

1

2 1 2 1

1

2 1

2 12 1

2( )( )

sinkkpm

x a k ma k m

π ππ2 12 2 1

2 21

++ +−tan

cos[ /( )]sin[ /( ++

⎛⎝⎜

⎞⎠⎟

− −+

=

−

− +

∑∫ 1

12 1

2

1

1

2 1

)]

( )( )

cos

k

m

p

m pm akppm

x axk

ma

k

m π π2 1

22

2 1

1

2 2

1 ++

++

⎛⎝⎜

⎞⎠⎟

+ −

=∑ ln cos

( )) ln( )( )

p

m p

x am a

−

− +

++

1

2 12 1

where 0 < p � 2m + 1.

17.16.14. x dxx a m a

kpm

p

m m m p

−

+ + − +−= −

+ +

1

2 1 2 1 2 1

22 1

22( )

sinπ11

2 2 12 2 1

1tancos[ /( )]

sin[ /( )]− − +

+⎛⎝

x a k ma k m

ππ⎜⎜

⎞⎠⎟

++ +

−

=

− +

∑∫k

m

m pm akpm

x

1

2 121

2 122 1

2( )

cos lnπ

aaxk

ma

x am a

k

m

cos

ln( )( )

22 1

2 1

2

1

2

π+

+⎛⎝⎜

⎞⎠⎟

+ −+

=∑

mm p− +1

where 0 < p � 2m + 1.


89

(17) Integrals Involving sin ax

17.17.1. sincos

ax dxax

a= −∫

17.17.2. x ax dxax

ax ax

asin

sin cos= −∫ 2

17.17.3. x ax dxx

aax

axa

ax22 3

22 2sin sin cos= + −⎛

⎝⎜⎞⎠⎟∫

17.17.4. x ax dxx

a aax

xa

xa

32

2 4 3

33 6 6sin sin= −⎛

⎝⎜⎞⎠⎟

+ −⎛⎝⎜

⎞⎠⎠⎟∫ cos ax

17.17.5. sin ( )

!( )

!ax

xdx ax

ax ax= −⋅

+⋅

− ⋅ ⋅ ⋅∫3 5

3 3 5 5

17.17.6. sin sin cos

(ax

xdx

axx

aax

xdx

2= − + ∫∫ See 17.18.5.)

17.17.7.dx

axax ax

axsin

ln(csc cot ) ln tan= − =∫1 1

2α α

17.17.8. x dx

ax aax

ax ax n

sin( ) ( ) (

= + + + +−1

1871800

2 22

3 5 2 1

�−−

++

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+

∫1

2 1

2 1) ( )

( )!

B ax

nn

n

�

17.17.9. sinsin2

22

4ax dx

x axa

= −∫

17.17.10. x ax dxx x ax

aax

asin

sin cos22

242

42

8= − −∫

17.17.11. sincos cos3

3

3ax dx

axa

axa∫ = − +

17.17.12. sinsin sin4 3

82

44

32∫ = − +ax dxx ax

aaxa

17.17.13.dx

ax aax

sincot2

1∫ = −

17.17.14.dx

axax

a ax aax

sincossin

ln tan3 221

2 2= − +∫

17.17.15. sin sinsin ( )

( )sin ( )

(px qx dx

p q xp q

p q xp q

= −−

− ++2 2 ))

( ,∫ = ±If see 17.17.9.)p q

17.17.16.dx

ax aax

11

4 2− = +⎛⎝⎜

⎞⎠⎟∫ sin

tanπ

17.17.17.xdx

axxa

axa

ax1 4 2

24 22−

= +⎛⎝⎜

⎞⎠⎟

+ −sin

tan ln sinπ π⎛⎛

⎝⎜⎞⎠⎟∫

17.17.18.dx

axax

11

4 2+ = − −⎛⎝⎜

⎞⎠⎟∫ sin

tanαπ

17.17.19.x dx

axxa

axa

ax1 4 2

242+ = − −⎛

⎝⎜⎞⎠⎟ + +

sintan ln sin

π π22

⎛⎝⎜

⎞⎠⎟∫


90

17.17.20.dx

ax aax

a( sin )tan tan

11

2 4 21

6 423

− = +⎛⎝⎜

⎞⎠⎟ + +∫

π π aax2

⎛⎝⎜

⎞⎠⎟

17.17.21.dx

ax aax

a( sin )tan tan

11

2 4 21

6 423

+= − −

⎛⎝⎜

⎞⎠⎟

−∫π π −−

⎛⎝⎜

⎞⎠⎟

ax2

17.17.22.dx

p q ax

a p q

p ax q

p q

a q

+ =−

+−

∫

−

sin

tantan2

1

2 2

112

2 2

2 −−+ − −+ + −

⎛

⎝⎜

⎞

⎠p

p ax q q p

p ax q q p2

12

2 2

12

2 2ln

tan

tan ⎟⎟

⎧

⎨

⎪⎪

⎩

⎪⎪

(If p = ± q, see 17.17.16 and 17.17.18.)

17.17.23.dx

p q axq ax

a p q p q axp

p( sin )cos

( )( sin )+ = − + + −2 2 2 2 qqdx

p q ax2 +∫∫ sin

(If p = ± q, see 17.17.20 and 17.17.21.)

17.17.24.dx

p q ax ap p q

p q axp2 2 2 2 2

12 21

+ =+

+−∫ sintan

tan

17.17.25.dx

p q ax

ap p q

p q axp

ap q

2 2 2

2 2

12 21

1

2

− =−

−−

sin

tantan

22 2

2 2

2 2−− +− −

⎛

⎝⎜

⎞

⎠⎟

⎧

⎨

⎪⎪

⎩

⎪p

q p ax p

q p ax pln

tan

tan⎪⎪

∫

17.17.26. x ax dxx ax

amx ax

am m

axm

m m

sincos sin ( )= − + − −∫

−1

2 2

1 mm ax dx−∫ 2 sin

17.17.27.sin sin

( )cosax

xdx

axn x

an

axx

dxn n n∫ ∫= −

−+

−− −1 11 1((See 17.18.30.)

17.17.28. sinsin cos

sinnn

nax dxax axan

nn

ax dx= − + −−−∫ ∫

121

17.17.29. dxax

axa n ax

nn

dxn n nsin

cos( )sin sin

= −− + −

−− −1211 22 ax∫∫

17.17.30. x dxax

x axa n ax a n nn nsin

cos( )sin ( )(

= −− − − −−1

111 2 22

212 2) sin sinn nax

nn

x dxax− −+ −

−∫ ∫

(18) Integrals Involving cos ax

17.18.1. cossin

ax dxax

a=∫

17.18.2. x ax dxax

ax ax

acos

cos sin= +∫ 2

17.18.3. x ax dxx

aax

xa a

ax22

2

3

2 2cos cos sin= + −⎛

⎝⎜⎞⎠⎟∫


91

17.18.4. x ax dxx

a aax

xa

xa

32

2 4

3

3

3 6 6cos cos= −⎛

⎝⎜⎞⎠⎟

+ −⎛⎝⎜

⎞⎠⎠⎟∫ sin ax

17.18.5. cosln

( )!

( )!

( )!

axx

dx xax ax ax= − ⋅ + ⋅ − ⋅ +

2 4 6

2 2 4 4 6 6��∫

17.18.6. cos cos sin

(ax

xdx

axx

aax

xdx

2= − − ∫∫ See 17.17.5.)

17.18.7. dx

ax aax ax

aax

cosln (sec tan ) ln tan= + = +⎛

⎝⎜⎞1 1

4 2π

⎠⎠⎟∫

17.18.8.x dx

ax aax ax ax E axn

cos( ) ( ) ( ) (

= + + + +12 8

51442

2 4 6

�))

( )( )!

2 2

2 2 2

n

n n

+

+ +⎧⎨⎩

⎫⎬⎭∫ �

17.18.9. cossin2

22

4ax dx

x axa

= +∫

17.18.10. x ax dxx x ax

aax

acos

sin cos22

242

42

8= + +∫

17.18.11. cossin sin3

3

3ax dx

axa

axa

= −∫

17.18.12. cossin sin4 3

82

44

32ax dx

x axa

axa

= + +∫

17.18.13. dxax

axacos

tan2 =∫

17.18.14.dx

axax

a ax aax

cossincos

ln tan3 221

2 4 2= + +⎛

⎝⎜⎞⎠⎟

π∫∫

17.18.15. cos cossin( )

( )sin( )

(ax px dx

a p xa p

a p xa∫ = −

−+ +

+2 2 ppa p

)( ,If see 17.18.9.)= ±

17.18.16.dx

ax aax

11

2− = −∫ coscot

17.18.17.x dx

axxa

axa

ax1 2

222− = − +∫ cos

cot ln sin

17.18.18.dx

ax aax

11

2+ =∫ costan

17.18.19.x dx

axxa

axa

ax1 2

222+ = +∫ cos

tan ln cos

17.18.20.dx

ax aax

aax

( cos )cot cot

11

2 21

6 223

− = − −∫

17.18.21.dx

ax aax

aax

( cos )tan tan

11

2 21

6 223

+ = +∫

17.18.22. dxp q ax

a p qp q p q ax

a

+=

−− +−

cos

tan ( ) / ( ) tan2 1

2

1

2 2

1

qq p

ax q p q p

ax q p q2 2

12

12−

+ + −

− +ln

tan ( ) / ( )

tan ( ) / ( −−

⎛

⎝⎜

⎞

⎠⎟

⎧

⎨

⎪⎪

⎩

⎪⎪

∫p)

( ,If see 17.18.16and 17.18.18.)

p q= ±


92

17.18.23.dx

p q axq ax

a q p p q axp

q( cos )sin

( )( cos )+ = − + − −2 2 2 2 ppdx

p q ax2 +∫∫ cos(If see 17.18.19and 17.18.20.)

p q= ±

17.18.24.dx

p q ax ap p q

p ax

p q2 2 2 2 2

1

2 2

1+ =

+ +−∫ cos

tantan

17.18.25.dx

p q ax

ap p q

p ax

p q

ap q

2 2 2

2 2

1

2 2

1

1

2

− =− −

−

cos

tantan

22 2

2 2

2 2−− −+ −

⎛

⎝⎜

⎞

⎠⎟

⎧

⎨⎪⎪

⎩⎪

p

p ax q p

p ax q pln

tan

tan⎪⎪

∫

17.18.26. x ax dxx ax

amx

aax

m ma

xmm m

m∫ = + − −−

cossin

cos( )1

2 2

1 −−∫ 2 cos ax dx

17.18.27.cos cos

( )sin

(ax

xdx

axn x

an

axx

dxn n n

= −−

−−− −1 11 1

Seee 17.17.27.)∫∫

17.18.28. cossin cos

cosnn

nax dxax ax

ann

nax dx= + −−

−∫ ∫1

21

17.18.29.dx

axax

a n axnb

dxn n ncos

sin( ) cos cos

= − + −−− −1

211 2 aax∫∫

17.18.30.x dx

axx ax

a n ax a n nn ncossin

( ) cos ( )(= − − − −−1

11 21 2 )) cos cosn nax

nn

x dxax− −+ −

−∫ ∫2 2

21

(19) Integrals Involving sin ax and cos ax

17.19.1. sin cossin

ax ax dxax

a=∫

2

2

17.19.2. sin coscos( )

( )cos( )

(px qx dx

p q xp q

p q xp

= − −− − +

+2 2 qq)

17.19.3. sin cossin( )

( ,nn

ax ax dxax

n an∫ =

+= −

+1

11If see 117.21.1.)

17.19.4. cos sincos( )

,nn

ax ax dxax

n an∫ = −

+= −

+1

11(If see 17.20.1.)

17.19.5. sin cossin2 2

84

32ax ax dx

x axa

= −∫

17.19.6.dx

ax ax aax

sin cosln tan=∫

1

17.19.7.dx

ax ax aax

a axsin cosln tan

sin2

14 2

1= +⎛⎝⎜

⎞⎠⎟ −∫

π

17.19.8.dx

ax ax aax

a axsin cosln tan

cos2

12

1= +∫

17.19.9.dx

ax axax

asin coscot

2 2

2 2= −∫


93

17.19.10.sincos

sinln tan

2 12 4

axax

dxax

a aax= − + +⎛

⎝⎜⎞⎠⎟∫

π

17.19.11.cossin

cosln tan

2 12

axax

dxax

a aax= +∫

17.19.12.dx

ax ax a ax aax

cos ( sin ) ( sin )ln tan

11

2 11

2 2± = ± +∫ ∓ ++⎛⎝⎜

⎞⎠⎟

π4

17.19.13.dx

ax ax a ax aax

sin ( cos ) ( cos )ln tan

11

2 11

2 2± = ± ± +∫

17.19.14.dx

ax ax a

axsin cos

ln tan± = ±⎛⎝⎜

⎞⎠⎟∫

1

2 2 8π

17.19.15.sin

sin cosln (sin cos )

ax dxax ax

xa

ax ax± = ±∫ 21

2∓

17.19.16.cos

sin cosln (sin cos )

ax dxax ax

xa

ax ax± = ± + ±∫ 21

2

17.19.17.sin

cosln ( cos )

ax dxp q ax aq

p q ax+ = − +∫1

17.19.18.cos

sinln ( sin )

ax dxp q ax aq

p q ax+ = +∫1

17.19.19.sin

( cos ) ( )( cos )ax dx

p q ax aq n p q axn n+ = − + −∫1

1 1

17.19.20.cos

( sin ) ( )( sin )ax dx

p q ax aq n p q axn n+ = −− + −∫

11 1

17.19.21.dx

p ax q ax a p q

ax q psin cos

ln tantan ( / )

+=

++⎛ −1

22 2

1

⎝⎝⎜⎞⎠⎟∫

17.19.22.dx

p ax q ax r

a r p q

p r q

sin cos

tan( ) tan

+ +=

− −+ −−2

2 2 2

1 (( / )

ln

ax

r p q

a p q r

p p q r

2

1

2 2 2

2 2 2

2 2 2

− −

⎛

⎝⎜

⎞

⎠⎟

+ −

− + − ++ −

+ + − + −

⎛

⎝

( ) tan ( / )

( ) tan ( / )

r q ax

p p q r r q ax

2

22 2 2⎜⎜

⎞

⎠⎟

⎧

⎨

⎪⎪

⎩

⎪⎪

∫

(If r = q see 17.19.23. If r2 = p2 + q2 see 17.19.24.)

17.19.23.dx

p ax q ax apq p

axsin ( cos )

ln tan+ + = +⎛⎝⎜

⎞⎠⎟∫ 1

12

17.19.24.dx

p ax q ax p q a p q

ax

sin costan

tan

+ ± += −

++ −

2 2 2 2

14π

∓11

2( / )q p⎛

⎝⎜⎞⎠⎟∫

17.19.25.dx

p ax q ax apqp ax

q2 2 2 211

sin costan

tan+ = ⎛

⎝⎜⎞⎠⎟

−∫

17.19.26.dx

p ax q ax apqp ax qp ax q2 2 2 2

12sin cos

lntantan− = −

+⎛⎛⎝⎜

⎞⎠⎟∫


94

17.19.27. sin cos

sin cos( )m n

m n

ax ax dx

ax axa m n

mm=

− + + −− +1 1 1++

+ +

−

+ −

∫nax ax dx

ax axa m n

m n

m n

sin cos

sin cos( )

2

1 1 nnm n

ax ax dxm n−+

⎧

⎨⎪

⎩⎪ −∫

∫ 1 2sin cos

17.19.28.sincos

sin( ) cos

m

n

m

n

axax

dx

axa n ax

mn

=

− − −−

−

−

1

11111

1

2

2

1

1

sincos

sin( ) cos

m

n

m

n

axax

dx

axa n ax

−

−

+

−

∫

− −− − +−

−−

−

−

∫m n

naxax

dx

axa m n

m

n

m

21 2

1

sincos

sin( ) coos

sincosn

m

naxmm n

axax

dx−

−

+ −−

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪ ∫

∫

1

21

17.19.29.cossin

cos( )sin

m

n

m

n

axax

dx

axa n ax

mn

=

−− − −−

−

1

111

−−−

−

−

−

+

−

∫1

1

2

2

1

1

cossin

cos( )sin

m

n

m

n

axax

dx

axa n aax

m nn

axax

dx

axa m n

m

n

m

− − +−

−

−

−

∫2

1 2

1

cossin

cos( )ssin

cossinn

m

naxmm n

axax

dx−

−

+ −−

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪ ∫

∫

1

21

17.19.30.dx

ax axa n ax ax

m n

m n

m n

sin cos( )sin cos= − + + −

− −1

1 1 1

221

11

2

1

ndx

ax ax

a m ax

m n

m n

−−

−

−

− −

∫ sin cos

( )sin cos 11 2

21ax

m nm

dxax axm n+ + −

−

⎧

⎨⎪

⎩⎪⎪ −∫

∫sin cos

(20) Integrals Involving tan ax

17.20.1. tan ln cos ln secax dxa

axa

ax= − =∫1 1

17.20.2. tantan2 ax dx

axa

x= −∫

17.20.3. tantan

ln cos32

21

ax dxax

a aax= +∫

17.20.4. tan sectan( )

nn

ax ax dxax

n a2

1

1= +

+

∫

17.20.5.sectan

ln tan2 1axax

dxa

ax=∫

17.20.6.dx

ax aax

tanln sin=∫

1

17.20.7. x ax dxa

ax ax ax n

tan( ) ( ) ( ) (

= + + + +13 15

2105

22

3 5 7 2

�22 1

2 1

2 2 1nn

nB axn

−+ +⎧

⎨⎩

⎫⎬⎭

+

∫) ( )

( )!�

17.20.8.tan ( ) ( ) ( ) (ax

xdx ax

ax ax Bn nn= + + + +

−3 5 2 2

92

752 2 1

�aax

n n

n)( )( )!

2 1

2 1 2

−

− +∫ �

17.20.9. x ax dxx ax

a aax

xtan

tanln cos2

2

212

= + −∫


95

17.20.10.dx

p q axpx

p qq

a p qq ax p a+ = + + + +

tan ( )ln ( sin cos2 2 2 2 xx)∫

17.20.11. tantan( )

tannn

nax dxax

n aax dx=

−−

−−∫∫

12

1

(21) Integrals Involving cot ax

17.21.1. cot ln sinax dxa

ax∫ = 1

17.21.2. cotcot2 ax dx

axa

x= − −∫

17.21.3. cotcot

ln sin32

21

∫ = − −ax dxax

a aax

17.21.4. cot csccot( )

nn

ax ax dxax

n a∫ = −+

+2

1

1

17.21.5. csccot

ln cot2 1axax

dxa

ax= −∫

17.21.6. dx

ax aax

cotln cos= −∫

1

17.21.7. x ax dxa

axax ax B axn

ncot( ) ( ) ( )

∫ = − − − −19 225

22

3 5 2

�22 1

2 1

n

n

+

+−

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪( )!�

17.21.8. cot ( ) ( )

(ax

xdx

axax ax B axn

nn

= − − − − −−1

3 135

23 2 2 1

�22 1 2n n−

−∫ )( )!�


a aax

xcot

cotln sin2

2

212∫ = − + −

17.21.10.dx

p q axpx

p qq

a p qq ax q a

+=

+−

++

cot ( )ln ( sin cos

2 2 2 2xx)∫

17.21.11. cotcot( )

cotnn

nax dxax

n aax dx∫ ∫= −

−−

−−

12

1

(22) Integrals Involving sec ax

17.22.1. sec ln (sec tan ) ln tanax dxa

ax axa

ax= + = +⎛⎝⎜

⎞1 12 4

π⎠⎠⎟∫

17.22.2. sectan2∫ =ax dx

axa

17.22.3. secsec tan

ln (sec tan )3

21

2∫ = + +ax dxax ax

a aax ax


96

17.22.4. sec tansecn

n

ax ax dxax

na∫ =

17.22.5. dx

axax

asecsin=∫

17.22.6. x ax dxa

ax ax ax E axnsec

( ) ( ) ( ) (= + + + +1

2 851442

2 4 6

�))

( )( )!

2 2

2 2 2

n

n n

+

++

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪∫ �

17.22.7.sec

ln( ) ( ) ( )ax

xdx x

ax ax ax= + + + + +2 4 6

45

9661

4320�

EE ax

n nn

n( )

( )!

2

2 2+∫ �

17.22.8. x ax dxxa

axa

axsec tan ln cos22

1∫ = +

17.22.9.dx

q p axxq

pq

dxp q ax+

= −+∫ ∫sec cos

17.22.10. secsec tan

( )secn

nnax dx

ax axa n

nn∫ ∫=

−+ −

−

−−

22

121

aax dx

(23) Integrals Involving csc ax

17.23.1. csc ln (csc cot ) ln tanax dxa

ax axa

ax= − =∫1 1

2

17.23.2. csccot2∫ = −ax dx

axa

17.23.3. csccsc cot

ln tan3

21

2 2∫ = − +ax dxax ax

a aax

17.23.4. csc cotcscn

n

ax ax dxax

na∫ = −

17.23.5. dx

axax

acsccos= −∫

17.23.6. x ax dxa

axax ax n

csc( ) ( ) (

= + + + +∫−1

1871800

2 22

3 5 2

�11 2 11

2 1

−+

+⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

+) ( )

( )!

B ax

nn

n

�

17.23.7. csc ( ) ( )ax

xdx

axax ax Bn

n= − + + + +−−1

671080

2 2 13 2 1

�(( )

( )( )!

ax

n n

n2 1

2 1 2

−

−+∫ �


a aaxcsc

cotln sin2

2

1∫ = − +

17.23.9. dx

q p axxq

pq

dxp q ax+

= −+∫ ∫csc sin

(See 17.17.22.)

17.23.10. csccsc cot

( )cscn

nnax dx

ax axa n

nn

= −−

+ −−

−−∫

22

121 ∫∫ ax dx


97

(24) Integrals Involving Inverse Trigonometric Functions

17.24.1. sin sin− −= + −∫ 1 1 2 2xa

dx xxa

a x

17.24.2. xxa

dxx a x

ax a x

sin sin− −= −⎛⎝⎜

⎞⎠⎟

+ −∫ 12 2

12 2

2 4 4

17.24.3. xxa

dxx x

ax a a x2 1

31

2 2 2 2

32

9∫ − −= + + −sin sin

( )

17.24.4.sin ( / ) ( / ) ( / )−

= + +1 3 5

2 3 31 32 4 5

x ax

dxxa

x a x ai i

ii i ii

i ii i i i

�5

1 3 52 4 6 7 7

7

+ +∫( / )x a

17.24.5.sin ( / ) sin ( / )

ln− −

= − − + −⎛

⎝∫1

2

1 2 21x ax

dxx a

x aa a x

x⎜⎜⎞

⎠⎟

17.24.6. sin sin− −⎛⎝⎜

⎞⎠⎟ = ⎛

⎝⎜⎞⎠⎟ − + −∫ 1

2

1

2

2 22 2xa

dx xxa

x a x ssin−1 xa

17.24.7. cos cos− −= − −∫ 1 1 2 2xa

dx xxa

a x

17.24.8. xxa

dxx a x

ax a x

cos cos− −∫ = −⎛⎝⎜

⎞⎠⎟

− −12 2

12 2

2 4 4

17.24.9. xxa

dxx x

ax a a x2 1

31

2 2 2 2

32

9∫ − −= − + −cos cos

( )

17.24.10.cos ( / )

lnsin ( / )

(− −

= − ∫∫1 1

2x a

xdx x

x ax

dxπ

See 17.224.4.)

17.24.11.cos ( / ) cos ( / )

ln− −

= − + + −⎛

⎝∫1

2

1 2 21x ax

dxx a

x aa a x

x⎜⎜⎞

⎠⎟

17.24.12. cos cos− −⎛⎝⎜

⎞⎠⎟ = ⎛

⎝⎜⎞⎠⎟ − − −∫ 1

2

1

2

2 22 2xa

dx xxa

x a x ccos−1 xa

17.24.13. tan tan ln ( )− −= − +∫ 1 1 2 2

2xa

dx xxa

ax a

17.24.14. xxa

dx x axa

axtan ( ) tan− −= + −∫ 1 2 2 11

2 2

17.24.15. xxa

dxx x

aax a

x a2 13

12 3

2 2

3 6 6tan tan ln ( )− −= − + +∫

17.24.16. tan ( / ) ( / ) ( / ) ( / )−

= − + −1 3

2

5

2

7

23 5 7x a

xdx

xa

x a x a x a ++∫ �

17.24.17.tan ( / )

tan ln−

−= − − +⎛⎝⎜

⎞⎠

1

21

2 2

2

1 12

x ax

dxx

xa a

x ax ⎟⎟∫


98

17.24.18. cot cot )− −∫ = + +1 1 2 2

2xa

dx xxa

ax aln (

17.24.19. xxa

dx x axa

axcot ( ) cot− −∫ = + +1 2 2 11

2 2

17.24.20. xxa

dxx x

aax a

x a2 13

12 3

2 2

3 6 6cot cot ( )− −∫ = + − +ln

17.24.21.cot ( / ) tan ( / )− −

∫ ∫= −1 1

2x a

xdx x

x ax

dxπ

ln (See 17.24.16.)

17.24.22.cot ( / ) cot ( / )− −

∫ = + +⎛⎝⎜

1

2

1 2 2

2

12

x ax

dxx a

x ax a

xln

⎞⎞⎠⎟

17.24.23. secsec ( ) sec

−

− −

=− + − < <

1

1 2 2 102x

adx

xxa

a x x axa

lnπ

xxxa

a x x axa

sec sec− −+ + − < <

⎧

⎨⎪

⎩⎪

∫1 2 2 1

2ln ( )

π π

17.24.24. xxa

dx

x xa

a x a xa

xsec

sec sec−

− −

=− − < <

1

21

2 21

2

2 20

2π

22 2 21

2 21sec sec− −+ − < <

⎧

⎨⎪⎪

⎩⎪⎪

∫xa

a x a xa

π π

17.24.25. xxa

dx

x xa

ax x a ax x

2 1

31

2 2 32

3 6 6secsec

−

−

=− − − + −ln( aa

xa

x xa

ax x a ax

2 1

31

2 2 3

02

3 6 6

) sec

sec

< <

+ − +

−

−

π

ln( ++ − < <

⎧

⎨⎪⎪

⎩⎪⎪

−∫

x axa

2 2 1

2) sec

π π

17.24.26.sec ( / ) ( / ) ( /−

= + + +∫1 3

2 2 3 31 3x a

xdx x

ax

a x a xπln

i ii )) ( / )5 7

2 4 5 51 3 52 4 6 7 7i i ii ii i i i

+ + ⋅⋅ ⋅a x

17.24.27.sec ( / )

sec ( / )sec

−

−−

=

− + − <1

2

1 2 210

x ax

dx

x ax

x aax

xaa

x ax

x aax

xa

<

− − − < <

⎧

⎨

⎪⎪⎪

⎩

−−

π

π π

2

2

1 2 21sec ( / )

sec⎪⎪⎪⎪

∫

17.24.28. csccsc ( ) csc

−

− −

=+ + − < <

1

1 2 2 102x

adx

xxa

a x x axa

lnπ

xxxa

a x x axa

csc ( ) csc− −− + − − < <

⎧

⎨⎪

⎩⎪

∫1 2 2 1

20ln

π

17.24.29. xxa

dx

x xa

a x a xa

xcsc

csc csc−

− −

=+ − < <

1

21

2 21

2

2 20

2π

22 2 201

2 21csc csc− −− − − < <

⎧

⎨⎪⎪

⎩⎪⎪

∫xa

a x a xa

π

17.24.30. xxa

dx

x xa

ax x a ax x

2 1

31

2 2 32

3 6 6csc

csc (−

−

=+ − + +

∫ln −− < <

− − −

−

−

axa

x xa

ax x a a

2 1

31

2 2 3

02

3 6 6

) csc

csc (

π

ln xx x axa

+ − − < <

⎧

⎨⎪⎪

⎩⎪⎪ −2 2 1

20) csc

π


99

17.24.31.csc ( ) ( ) ( )−

∫ = − + +1 3 51 3

4x a

xdx

ax

a x a x/ /2 3 3

/2i iii ii i

i ii i i i5 5

/2 6 7 7

+ + ⋅⋅ ⋅⎛⎝⎜

⎞⎠⎟

1 3 54

7( )a x

17.24.32.csc ( )

csc ( )csc

−

−−

∫ =− − − <

1

2

1 2 210

x ax

dx

x ax

x aax/

/ xxa

x ax

x aax

xa

<

− + − − < <

⎧

⎨⎪⎪

−−

π

π

2

20

1 2 21csc ( )

csc/

⎩⎩⎪⎪

17.24.33. xxa

dxxm

xa m

x

a xdxm

m m

sin sin−+

−+

= + − + −∫11

11

2 211

1∫∫

17.24.34. xxa

dxxm

xa m

x

a xdxm

m m

cos cos−+

−+

=+

++ −∫ 1

11

1

2 211

1 ∫∫

17.24.35. xxa

dxxm

xa

am

xx a

dxmm m

tan tan−+

−+

=+

−+ +∫ 1

11

1

2 21 1 ∫∫

17.24.36. xxa

dxxm

xa

am

xx a

dxmm m

cot cot−+

−+

=+

++ +∫ 1

11

1

2 21 1 ∫∫

17.24.37. xxa

dx

x x am

am

x dx

x am

m m

sec

sec ( / )

−

+ −

=+

−+ −

∫ 1

1 1

21 1 22

1

1 1

2

02

1 1

∫ < <

++

+

−

+ −

sec

sec ( / )

xa

x x am

am

x dx

x

m m

π

−−< <

⎧

⎨⎪⎪

⎩⎪⎪ ∫ −

a

xa2

1

2π πsec

17.24.38. xxa

dx

x x am

am

x dx

x am

m m

csc

sec ( )

−

+ −

=+

++ −

∫ 1

1 1

21 1/

22

1

1 1

2

02

1 1

∫ < <

+−

+

−

+ −

csc

csc ( )

xa

x x am

am

x dx

x

m m

π

/

−−− < <

⎧

⎨⎪⎪

⎩⎪⎪ ∫ −

a

xa2

1

20

πcsc

(25) Integrals Involving eax

17.25.1. e dxea

axax

∫ =

17.25.2. xe dxea

xa

axax

∫ = −⎛⎝⎜

⎞⎠⎟

1

17.25.3. x e dxea

xx

a aax

ax2 2

2

2 2∫ = − +

⎛⎝⎜

⎞⎠⎟

17.25.4. x e dxx e

ana

x e dx

ea

xnx

a

n axn ax

n ax

axn

n

∫ ∫= −

= − +

−

−

1

1 nn n xa

na

nn n

n

( ) ( ) !− − ⋅⋅ ⋅ −⎛⎝⎜

⎞⎠⎟

=−1 12

2 if positivee integer

17.25.5.ex

dx xax ax axax

= + + + + ⋅⋅ ⋅∫ ln1 1 2 2 3 3

2 3

i i i!( )

!( )

!

17.25.6.ex

dxe

n xa

nex

dxax

n

ax

n

ax

n∫ ∫= −− + −− −( )1 11 1


100

17.25.7. dx

p qexp ap

p qeax

ax

+= − +∫

1ln ( )

17.25.8. dx

p qexp ap p qe ap

p qeax ax

ax

( ) ( ))

+= +

+− +∫ 2 2 2

1 1ln (

17.25.9.dx

pe qe

a pq

pq

e

a pq

eax ax

ax

+ =

⎛⎝⎜

⎞⎠⎟

−

−

−

∫

1

1

2

1tan

lnaax

ax

q p

e q p

− −+ −

⎛

⎝⎜⎞

⎠⎟

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

/

/

17.25.10. e bx dxe a bx b bx

a bax

ax

sin( sin cos )= −

+∫ 2 2

17.25.11. e bx dxe a bx b bx

a bax

ax

cos( cos sin )= +

+∫ 2 2

17.25.12. xe bx dxxe a bx b bx

a beax

ax ax

sin( sin cos ) {(= −

+−∫ 2 2

aa b bx ab bxa b

2 2

2 2 2

2− −+

) sin cos }( )

17.25.13. xe bx dxxe a bx b bx

a beax

ax ax

cos( cos sin ) {(= +

+ −∫ 2 2

aa b bx ab bxa b

2 2

2 2 2

2− ++

) cos sin }( )

17.25.14. e x dxe x

a aex

dxaxax ax

lnln∫ ∫= − 1

17.25.15. e bx dxe bxa n b

a bx nbax nax n

∫ = + −−

sinsin

( sin co1

2 2 2 ss )( )

sinbxn n ba n b

e bx dxax n+ −+

−∫1 2

2 2 22

17.25.16. e bx dxe bx

a n ba bx nbax n

ax n

∫ = + +−

coscos

( cos si1

2 2 2 nn )( )

cosbxn n ba n b

e bx dxax n+ −+

−∫1 2

2 2 22

(26) Integrals Involving ln x

17.26.1. ln lnx dx x x x= −∫

17.26.2. x x dxx

xln ln= −⎛⎝⎜

⎞⎠⎟∫

2

212

17.26.3. x x dxxm

xm

mmm

ln ln If see=+

−+

⎛⎝⎜

⎞⎠⎟

= −+

∫1

11

11 17( , .. . .)26 4

17.26.4. lnln

xx

dx x=∫12

2

17.26.5.ln lnxx

dxx

x x2

1= − −∫17.26.6. ln ln ln2 2 2 2x dx x x x x x∫ = − +

17.26.7.ln ln

If seen nx dxx

xn

n=+

= −+

∫1

11 17 26 8( , . . .)

17.26.8.dx

x xx

lnln ln=∫ ( )


101

17.26.9. dx

xx x

x xln

ln ln= + + + + ⋅⋅ ⋅∫ ( ) lnln

!ln

!

2 3

2 2 3 3i i

17.26.10.x dx

xx m x

m x mm

lnln ln= + + + + + +

( ) ( ) ln( ) ln

!(

11

2 2

2 2

i11

3 3

3 3) ln!

xi∫ + ⋅ ⋅ ⋅

17.26.11. ln lnn n nx dx x x n x dx= − −∫∫ ln 1

17.26.12. x x dxx x

mn

mx xdxm n

m nm nln

lnln=

+−

+

+−∫ ∫

11

1 1

If m = –1, see 17.26.7.

17.26.13. ln ( ln (x a dx x x a x axa

2 2 2 2 12 2+ = + − + −∫ ) ) tan

17.26.14. ln ( ) ln ( ) lnx a dx x x a x ax ax a

2 2 2 2 2− = − − + +−

⎛⎝⎜

⎞⎠⎟∫

17.26.15. x x a dxx x a

m mx

x am

m m

ln )ln ( )

( 2 21 2 2 2

212

1± = ±

+ − + ±+ +

22 dx∫∫

(27) Integrals Involving sinh ax

17.27.1. sinhcosh

ax dxax

a∫ =


aax

asinh

cosh sinh∫ = −2

17.27.3. x ax dxxa a

axx

aax2

2

3 2

2 2sinh cosh sinh∫ = +

⎛⎝⎜

⎞⎠⎟

−

17.27.4.sinh ( )

!( )

!ax

xdx ax

ax ax= + + + ⋅⋅ ⋅∫3 5

3 3 5 5i i

17.27.5.sinh sinh coshax

xdx

axx

aax

xdx2 = − +∫ ∫

(See 17.28.4.)

17.27.6.dx

ax aax

sinhln tanh=∫

12

17.27.7.x dx

ax aax

ax axsinh

( ) ( ) (= − + − ⋅⋅ ⋅ +

−∫

118

71800

22

3 5 11 2 12 1

2 2 1) ( ) ( )( )!

n nn

nB axn−

+ + ⋅⋅ ⋅⎧⎨⎩

⎫⎬⎭

+

17.27.8. sinhsinh cos2

2 2ax dx

ax axa

x= −∫h


aax

ax

sinhsinh cosh2

2

224

28 4

= − −∫


102

17.27.10.dx

axax

asinhcoth

2= −∫

17.27.11. sinh sinhsinh ( )

( )sinh ( )

ax px dxa p x

a pa p x= +

+− −

∫ 2 22( )a p−

For a = ± p see 17.27.8.

17.27.12. x ax dxx ax

ama

x ax dxmm

msinhcosh

cosh= −∫ ∫ −1 (See 17.28.12.)

17.27.13. sinhsinh cosh

sinhnn

nax dxax axan

nn

ax d= − −∫

−−

121

xx∫

17.27.14. sinh sinh( )

coshaxx

dxax

n xa

nax

xn n n∫ = −−

+−− −1 11 1

ddx∫ (See 17.28.14.)

17.27.15.dx

axax

a n axnn

dxn nsinh

cosh( )sinh s∫ = −

−− −

−−1211 iinhn ax−∫ 2

17.27.16.x dx

axx ax

a n ax a nn nsinhcosh

( )sinh (∫ = −−

−−−1

111 2 ))( )sinh sinhn ax

nn

x dxaxn n−

− −−− −∫2

212 2

(28) Integrals Involving cosh ax

17.28.1. coshsinh

ax dxax

a∫ =


aax

acosh

sinh cosh∫ = −2


axa a

ax22

2

3

2 2cosh

coshsinh∫ = − + +

⎛⎝⎜

⎞⎠⎟

17.28.4.cosh

ln( )

!( )

!( )ax

xdx x

ax ax ax= + + +∫2 4 6

2 2 4 4 6 6i i i !!+ ⋅⋅ ⋅

17.28.5.cosh cosh sinhax

xdx

axx

aax

xdx2 = − +∫ ∫ (See 17.27.4.)

17.28.6.dx

ax aeax

coshtan= −∫

2 1

17.28.7.x dx

ax aax ax ax

cosh( ) ( ) ( ) (

= − + + ⋅⋅ ⋅ +∫1

2 851442

2 4 6 −−+ + ⋅⋅ ⋅⎧

⎨⎩

⎫⎬⎭

+12 2 2

2 2) ( )( )( )!

nn

nE axn n

17.28.8. coshsinh cosh2

2 2ax dx

x ax axa

= +∫

17.28.9. x ax dxx x ax

aax

acosh

sinh cosh22

242

42

8= + −∫


103

17.28.10.dx

axax

acoshtanh

2=∫

17.28.11. cosh coshsinh( )

( )sinh( )

ax px dxa p xa p

a p x= −−

+ +∫ 2 22( )a p+

17.28.12. x ax dxx ax

ama

x ax dxmm

mcoshsinh

sinh= −∫ ∫ −1 (See 17.27.12.)

17.28.13. coshcosh sinh

coshnn

nax dxax axan

nn

ax d= + −∫−

−1

21xx∫

17.28.14.cosh cosh

( )sinhax

xdx

axn x

an

axxn n n∫ = −

−+

−− −1 11 1ddx∫ (See 17.27.14.)

17.28.15.dx

axax

a n axnn

dxn ncosh

sinh( ) cosh co∫ =

−+ −

−−1211 sshn ax−∫ 2

17.28.16.x dx

axx ax

a n ax n nn ncoshsinh

( ) cosh ( ) (∫ =−

+−−1

111 −−

+ −−− −∫2

212 2 2) cosh cosha ax

nn

x dxaxn n

(29) Integrals Involving sinh ax and cosh ax

17.29.1. sinh coshsinh

ax ax dxax

a=∫

2

2

17.29.2. sinh coshcosh ( )

( )cosh ( )

px qx dxp q x

p qp q x= +

++ −

∫ 2 22( )p q−

17.29.3. sinh coshsinh2 2 4

32 8ax ax dx

axa

x= −∫

17.29.4.dx

ax ax aax

sinh coshln tanh∫ = 1

17.29.5.dx

ax axax

asinh coshcoth

2 2

2 2∫ = −

17.29.6. sinhcosh

sinhtan sinh

211ax

axdx

axa a

ax∫ = −

17.29.7.coshsinh

coshln tanh

2 12

axax

dxax

a aax∫ = +


104

(30) Integrals Involving tanh ax

17.30.1. tanh ln coshax dxa

ax=∫1

17.30.2. tanhtanh2 ax dx x

axa

= −∫

17.30.3. tanh ln coshtanh3

212

ax dxa

axax

a= −∫

17.30.4. x ax dxa

ax ax axtanh

( ) ( ) ( ) (= − + − ⋅⋅ ⋅

−13 15

21052

3 5 7 11 2 2 1

2 1

1 2 2 2 1) ( ) ( )

( )!

n n nn

nB ax

n

− +−+

+ ⋅ ⋅ ⋅⎧⎨⎪

⎩⎪

⎫⎬⎬⎪

⎭⎪∫


a aaxtanh

tanhln cosh2

2

221= − +∫

17.30.6.tanh ( ) ( ) ( ) (ax

xdx ax

ax ax n n

= − + − ⋅⋅ ⋅− −3 5 1 2

92

75

1 2 22 1

2 1 2

2 2 1nn

nB ax

n n

−−

+ ⋅⋅ ⋅−

∫) ( )

( )( )!

17.30.7.dx

p q axpx

p qq

a p qq ax p+ = − − − +∫ tanh ( )

ln ( sinh c2 2 2 2 oosh )ax

17.30.8. tanhtanh

( ) tanhnn

nax dxax

a aax dx= −

− +−

−∫∫1

2

1

(31) Integrals Involving coth ax

17.31.1. coth ln sinhax dxa

ax∫ = 1

17.31.2. cothcoth2 ax dx x

axa∫ = −

17.31.3. coth ln sinhcoth3

212

ax dxa

axax

a∫ = −

17.31.4. x ax dxa

axax ax n

coth( ) ( ) ( )

∫ = + − + ⋅⋅ ⋅− −1

9 2251

2

3 5 1222 1

2 2 1nn

nB axn

( )( )!

+

+ + ⋅ ⋅ ⋅⎧⎨⎩

⎫⎬⎭


a aaxcoth

cothln sinh2

2

221∫ = − +

17.31.6.coth ( ) ( ) (ax

xdx

axax ax B an n

n= − + − + ⋅⋅ ⋅−1

3 1351 23 2 xx

n n

n)( )( )!

2 1

2 1 2

−

− + ⋅ ⋅ ⋅∫

17.31.7.dx

p q axpx

p qq

a p qp ax q+ = − − − +∫ coth ( )

ln ( sinh c2 2 2 2 oosh )ax

17.31.8. cothcoth

( )cothn

nnax dx

axa n

ax dx= − − +−

−∫∫1

2

1


105

(32) Integrals Involving sech ax

17.32.1. sech ax dxa

eax=∫ −2 1tan

17.32.2. sech2 ax dxax

a=∫

tanh

17.32.3. sech3 ax dxax ax

a aax= +∫ −sech tanh

tan sinh2

12

1

17.32.4. x ax dxa

ax ax axsech∫ = − + + ⋅⋅ ⋅

−12 8

51442

2 4 6( ) ( ) ( ) ( 112 2 2

2 2) ( )( )( )!

nn

nE axn n

+

+ + ⋅ ⋅ ⋅⎧⎨⎩

⎫⎬⎭


a aaxsech2 = −∫

tanhln cosh

12

17.32.6.sech

ln( ) ( ) ( )ax

xdx x

ax ax ax= − + − + ⋅2 4 6

45

9661

4320⋅⋅ ⋅

−+ ⋅ ⋅ ⋅∫

( ) ( )( )!

12 2

2nn

nE axn n

17.32.7. sech sechnn

ax dxax ax

a nnn

= − + −−∫

−sech tanh( )

2

121

nn ax dx−∫ 2

(33) Integrals Involving csch ax

17.33.1. csch ax dxa

ax=∫1

2ln tanh

17.33.2. csch2 ax dxax

a= −∫

coth

17.33.3. csch3 ax dxax ax

a aax= − −∫

csch cothln tanh

21

2 2

17.33.4. x ax dxa

axax ax

csch = − + + ⋅⋅ ⋅ +−

∫1

1871800

22

3 5( ) ( ) ( 11 2 12 1

2 1 2 1) ( ) ( )( )!

n nn

nB axn

− +−+ + ⋅ ⋅ ⋅⎧

⎨⎩

⎫⎬⎭


a aaxcsch2 = − +∫

cothlnsinh

12

17.33.6.csch ( ) ( ) (ax

xdx

axax ax n n

= − − + + ⋅⋅ ⋅−1

671080

1 2 23 2 −− −−− + ⋅ ⋅ ⋅∫1 2 11

2 1 2) ( )

( ) ( )!B ax

n nn

n

17.33.7. csch cscnn

ax dxax ax

a nnn

= −− − −

−∫−csch coth( )

2

121

hh 2n ax dx−∫


106

(34) Integrals Involving Inverse Hyperbolic Functions

17.34.1. sinh sinh− −∫ = − +1 1 2 2xa

dx xxa

x a

17.34.2. xxa

dxx a x

ax

x asinh sinh− −∫ = +⎛

⎝⎜⎞⎠⎟

− +12 2

12 2

2 4 4

17.34.3.sinh ( )

( ) ( )

−

∫ =

− +

1

3 5

2 3 31 32 4

x ax

dx

xa

x a x a

/

/ /i i

ii ii i

i ii i i i5 5

1 3 52 4 6 7 7

2

7

2

− + ⋅⋅ ⋅( )

ln (

x ax a

x a

/| | <

/ )) ( ) ( ) ( )2 2 2 2

1 32 4 4 4

1 3 52 4

− + −a x a x a x/ / /i i

ii i i

i i 66

2 2

2 4 6 6 6

22 2 2 2

i i i i

i i

+ ⋅⋅ ⋅

− − + −

x a

x a a x

>

/ /ln ( ) ( ) 11 32 4 4 4

1 3 52 4 6 6 6

4 6ii i i

i ii i i i

( ) ( )a x a xx

/ /+ − ⋅⋅ ⋅ << −

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

a

17.34.4. coshcosh ( ) , cosh ( )

−

− −

∫ =− − >

1

1 2 2 1xa

dxx x a x a x a/ / 00

01 2 2 1x x a x a x acosh ( ) , cosh ( )− −+ − <

⎧⎨⎪

⎩⎪ / /

17.34.5. xxa

dxx a x a x x a

cosh( )cosh ( )

−

−

∫ =− − −

1

2 2 1 2 214 2

14/ ,, cosh ( )

( )cosh ( )

−

−

>

− + −

1

2 2 1 2

0

14 2

14

x a

x a x a x x

/

/ aa x a2 1 0, cosh ( )− <

⎧

⎨⎪⎪

⎩⎪⎪ /

17.34.6.

cosh ( )ln ( )

( )−

∫ = ± + +1

221

22

1x ax

dx x aa x/

//

2 2 2i ii33 1 3 54 6( ) ( )a x a x/

2 4 4 4/

2 4 6 6 6i i ii ii i i i+ + ⋅⋅ ⋅⎡

⎣⎢⎤⎤⎦⎥

+ > − <− −if / if /cosh ( ) , cosh ( )1 10 0x a x a

17.34.7. tanh tanh ln( )− −∫ = + −1 1 2 2

2xa

dx xxa

aa x

17.34.8. xxa

dxax

x axa

tanh ( ) tanh− −∫ = + −1 2 2 1

212

17.34.9.tanh ( ) ( ) ( )−

∫ = + + + ⋅⋅ ⋅1 3

2

5

23 5x a

xdx

xa

x a x a/ / /

17.34.10. coth coth ln ( )− −∫ = + −1 1 2 2

2xa

dx x xa

x a

17.34.11. xxa

dxax

x axa

coth ( )coth− −∫ = + −1 2 2 1

212

17.34.12.coth ( ) ( ) ( )−

∫ = − + + + ⋅⋅ ⋅⎛⎝

1 3

2

5

23 5x a

xdx

ax

a x a x/ / /⎜⎜

⎞⎠⎟

17.34.13. sechsech ( ) sin ( ), sech

−

− − −

∫ =+

1

1 1 1xa

dxx x a a x a/ / (( )

sech ( ) sin ( ), sech (

x a

x x a a x a x a

/

/ / /

>

−− − −

0

1 1 1 )) <

⎧⎨⎪

⎩⎪ 0

17.34.14. csch sinh ( ,− − −∫ = ± + > −1 1 1 0xa

dx xxa

axa

x xcsch if if << 0)


107

17.34.15. xxa

dxxm

xa m

x

x am

m m

sinh sinh−+

−+

=+

−+ +∫ 1

11

1

2 211

1ddx∫

17.34.16. xxa

dx

xm

xa m

x

x am

m m

cosh

cosh

−

+−

+

∫ =+

−+ −

1

11

1

2 211

1ddx x a

xm

xa m

x

x

m m

∫ −

+−

+

>

++

+

cosh ( )

cosh

1

11

1

2

0

11

1

/

−−<

⎧

⎨⎪⎪

⎩⎪⎪ ∫ −

adx x a

2

1 0cosh ( )/

17.34.17. xxa

dxxm

xa

am

xa x

mm m

tanh tanh−+

−+

= + − + −∫ 11

11

2 21 1ddx∫

17.34.18. xxa

dxxm

xa m

xa x

mm m

coth coth−+

−+

= + − + −∫ 11

11

2 211

1ddx∫

17.34.19. xxa

dx

xm

xa

am

x dx

a xm

m m

sech

sech

−

+−

∫ =+

++ −

1

11

2 21 1 ∫∫ −

+−

>

+−

+ −

sech ( )

sech

1

11

2

0

11

1

x a

xm

xa m

x dx

a x

m m

/

22

1 0∫ − <

⎧

⎨⎪⎪

⎩⎪⎪ sech ( )x a/

17.34.20. xxa

dxxm

xa

am

x dx

x am

m m

csch csch−+

−= + ± + +∫ 11

1

2 21 1 ∫∫ + > − <( , )if ifx x0 0


18 DEFINITE INTEGRALS

Definition of a Definite Integral

Let f(x) be defined in an interval a � x � b. Divide the interval into n equal parts of length �x = (b − a)/n. Then the definite integral of f(x) between x = a and x = b is defined as

18.1. f x dx f a x f a x x f a xna

b( ) lim{ ( ) ( ) ( )= + + + +

→∞∫ Δ Δ Δ Δ Δ2 xx f a n x x+ + + −� ( ( ) ) }1 Δ Δ

The limit will certainly exist if f(x) is piecewise continuous.

If f xddx

g x( ) ( ),= then by the fundamental theorem of the integral calculus the above definite integral can

be evaluated by using the result

18.2. f x dxddx

g x dx g x g b g aa

b

a

b

a

b( ) ( ) ( ) ( ) ( )= = = −∫ ∫

If the interval is infinite or if f(x) has a singularity at some point in the interval, the definite integral is called an improper integral and can be defined by using appropriate limiting procedures. For example,

18.3. f x dx f x dxba a

b( ) lim ( )=

→∞

∞

∫ ∫

18.4. f x dx f x dxa a

b

b

( ) lim ( )=→−∞−∞

∞

→∞∫ ∫

18.5. f x dx f x dx b( ) lim ( )=→∈ 0

if is a singular point..α

∈b

a

b −

∫∫

18.6. f x dx dx aa

b

a

b( ) lim=

→ +∫∫ ∈ ∈0f x( ) if is a singulaar point.

General Formulas Involving Definite Integrals

18.7. { ( ) ( ) ( ) } ( ) ( )f x g x h x dx f x dx g x dxa

b

a

b

a

b± ± ± = ± ±∫ ∫� ∫∫ ∫ ±h x dx

a

b( ) �

18.8. cf x dx c f x dx ca

b

a( ) ( )=∫ where is any constant.

bb

∫

18.9. f x dxa

a( ) =∫ 0

18.10. f x dx f x dxa

b

b

a( ) ( )= −∫ ∫

18.11. f x dx f x dx f x dxa

b

a

c

c

b( ) ( ) ( )= +∫ ∫ ∫

108

109

18.12. f x dx b a f c c a ba

b( ) ( ) ( ) .= −∫ where is between and

This is called the mean value theorem for definite integrals and is valid if f(x) is continuous in a � x � b.

18.13. f x g x dx f c g x dx c aa

b( ) ( ) ( ) ( )=∫ where is between and bb

a

b

∫This is a generalization of 18.12 and is valid if f(x) and g(x) are continuous in a � x � b and g(x) � 0.

Leibnitz’s Rules for Differentiation of Integrals

18.14.d

dF x dx

Fd

dx Fddα α α φ α φ

αφ α

φ α

φ( , ) ( , )

( )

( )= ∂ +∫

1

2

22

11

2

11

( )

( )( , )

α

φ αφ α φ

α∫ − Fdd

Approximate Formulas for Definite Integrals

In the following the interval from x = a to x = b is subdivided into n equal parts by the pointsa = x0, x1, x2, …, xn–1, xn = b and we let y0 = f(x0), y1 = f(x1), y2 = f(x2), …, yn = f(xn), h = (b – a)/n.

Rectangular formula:

18.15. f x dx h y y y yna

b( ) ( )≈ + + + + −∫ 0 1 2 1�

Trapezoidal formula:

18.16. f x dxh

y y y y yn na

b( ) ( )≈ + + + + +−∫ 2

2 2 20 1 2 1�

Simpson’s formula (or parabolic formula) for n even:

18.17. f x dxh

y y y y y y yn n na( ) ( )≈ + + + + + + +− −3

4 2 4 2 40 1 2 3 2 1�bb

∫

Definite Integrals Involving Rational or Irrational Expressions

18.18.dx

x a a2 20 2+ =∞

∫π

18.19.x dx

x pp

p−∞

+ = < <∫1

0 10 1

ππsin

,

18.20. x dxx a

an m n

m nm

n n

m n

+ = + < + <∞ + −

∫0

1

10 1

ππsin[( ) / ]

,

18.21.x dx

x x mmm

1 2 20 + + =∞

∫ cos sinsinsinβ

ππ

ββ

18.22.dx

a x

a

2 20 2−=∫

π

18.23. a x dxaa

2 2

0

2

4− =∫

π

DEFINITE INTEGRALS

110

18.24. x a x dxa m n p

n mm n n p

m np

( )[( )/ ] ( )

[(− = + +

++ +1 1 1

1Γ Γ

Γ ))/ ]n pa

+ +∫ 10

18.25.x dx

x aa m n

n

m

n n r

r m nr

( )( ) [( )/ ]

sin[+ = − +− + −1 11 1π Γ(( ) / ]( )! [( ) / ]

,m n r m n r

m nr+ − + − + < + <∞

∫ 1 1 1 10 1

0 π Γ

Definite Integrals Involving Trigonometric Functions

All letters are considered positive unless otherwise indicated.

18.26. sin sin,

/ ,mx nx dx

m n m n

m n=

≠0

2

integers and

integeπ rrs and m n=

⎧⎨⎪

⎩⎪∫0

π

18.27. cos cos,

/ ,mx nx dx

m n m n

m n0

0

2

π

π∫ =

≠integers and

inntegers and m n=

⎧⎨⎪

⎩⎪

18.28. sin cos,

/ (mx nx dx

m n m n

m m=

+0

2 2

integers and even

−− +

⎧⎨⎪

⎩⎪∫

n m n m n20 ) , integers and odd

π

18.29. sin cos//

2 2

0

2

0

2

4x dx x dx= =∫∫

πππ

18.30. sin cos/

2 2

0

2 1 3 5 2 12 4 6 2 2

m mx dx x dxm

m= = −∫

π πi i �i i � ,, , ,

/m =∫ 1 2

0

2…

π

18.31. sin cos/

2 1

0

22 1 2 4 6 2

1 3 5 2m mx dx x dx

mm

+ +∫ = =π i i �

i i � ++ =∫ 11 2

0

2π /, , ,m …

18.32. sin cos( ) ( )( )

/2 1 2 1

0

2

2p qx x dx

p qp q

− − = +∫Γ ΓΓ

π

18.33. sin/

/

pxx

dx

p

p

p

=

>

=

− <

⎧

⎨⎪⎪

⎩⎪⎪

∞

∫π

π

2 0

0 0

2 0

0

18.34.sin cos

/

/

px qxx

dx

p q

p q

p q

0

0 0

2 0

4 0

∞

∫ =

> >

< <

= >

⎧

⎨⎪⎪π

π⎩⎩⎪⎪

18.35.sin sin /

/

px qxx

dxp p q

q p q20

2 0

2 0=

<

>

⎧⎨⎪

⎩⎪

∞

∫π

π

�

�

18.36.sin2

20 2px

xdx

p∞

∫ = π

18.37.1

220

− =∞

∫cos pxx

dxpπ

DEFINITE INTEGRALS

111

18.38.cos cos

lnpx qx

xdx

qp

− =∞

∫0

18.39. cos cos ( )px qxx

dxq p− = −∞

∫ 20 2π

18.40.cos mxx a

dxa

e ma2 20 2+ = −

∞

∫π

18.41.x mxx a

dx e masin2 20 2+ = −

∞

∫π

18.42.sin( )

( )mx

x x adx

ae ma

2 2 20 21+ = − −

∞

∫π

18.43.dx

a b x a b+ =−∫ sin

22 20

2 ππ

18.44.dx

a b x a b+ =−∫ cos

22 20

2 ππ

18.45.dx

a b xb a

a b+ =−∫

−

coscos ( / )/

0

2 1

2 2

π

18.46.dx

a b xdx

a b xa

a b( sin ) ( cos ) ( ) /+ = + = −∫ 20

2

2 2 2 3 2

2π π00

2π

∫

18.47.dx

a x a aa

1 22

10 12 20

2

− + = − < <∫ cos,

ππ

18.48.x x dxa x a

a a asincos

( / ) ln ( ),

ln (1 2

1 1

120 − + =

+ <∫

π π

π ++ >

⎧⎨⎪

⎩⎪ 1 1/ ),a a

18.49.cos

cos, , , , ,

mx dxa x a

aa

a mm

1 2 11 0 1 22 2

2

0 − + = − < =ππ…∫∫

18.50. sin cosax dx ax dxa

2 2

00

12 2

= =∞∞

∫∫π

18.51. sin ( / ) sin ,/ax dxna

nn

nnn= >

∞

∫1

12

110Γ π

18.52. cos ( / ) cos ,/ax dxna

nn

nnn= >

∞

∫1

12

110Γ π

18.53.sin cosx

xdx

x

xdx= =

∞∞

∫∫ 00 2π

18.54.sin

( )sin ( / ),

xx

dxp p

pp = < <∞

∫π

π2 20 1

0 Γ

18.55.cos

( ) cos ( / ),

xx

dxp p

pp = < <∞

∫π

π2 20 1

0 Γ

18.56. sin cos cos sinax bx dxa

ba

ba

2

0

2 2

212 2

∞

∫ = −⎛⎝⎜

⎞⎠⎟

π

DEFINITE INTEGRALS

112

18.57. cos cos cos sinax bx dxa

ba

ba

22 2

02

12 2

= +⎛⎝⎜

⎞⎠⎟

∞

∫π

18.58.sin3

30

38

xx

dx =∞

∫π

18.59.sin4

40 3x

xdx =

∞

∫π

18.60.tan x

xdx =

∞

∫π20

18.61.dx

xm1 40

2

+ =∫ tan/ ππ

18.62. xx

dxsin

/= − + − +{ }∫ 2

11

13

15

172 2 2 20

2�

π

18.63. tan−

∫ = − + − +1

0

1

2 2 2 2

11

13

15

17

xx

dx �

18.64.sin

ln−

∫ =1

0

1

22

xx

dxπ

18.65.1

0

1

1

− − =∫ ∫∞cos cosx

xdx

xx

dx γ

18.66.1

1 20 +−

⎛⎝⎜

⎞⎠⎟

=∞

∫ xx

dxx

cos γ

18.67.tan tan

ln− −∞ − =∫

1 1

0 2px qx

xdx

pq

π

Definite Integrals Involving Exponential Functions

Some integrals contain Euler’s constant g = 0.5772156 . . . (see 1.3, page 3).

18.68. e bx dxa

a bax−

∞= +∫ cos 2 20

18.69. e bx dxb

a bax−

∞= +∫ sin 2 20

18.70. e bxx

dxba

ax−∞−∫ =sin

tan0

1

18.71. e ex

dxba

ax bx− −∞ − =∫0ln

18.72. e dxa

ax−∞

=∫ 2 120

π

18.73. e bx dxa

eax b a− −∞

=∫ 2 212

4

0cos /π

DEFINITE INTEGRALS

113

18.74. e dxa

eb

aax bx c b ac a− + + −

∞=∫ ( ) ( )/2 21

2 24 4

0

πerfc

where erfc (p) = −∞

∫2 2

πe dxx

p

18.75. e dxa

eax bx c b ac a− + + −−∞

∞=∫ ( ) ( )/2 2 4 4π

18.76. x e dxna

n axn

−+

∞= +∫

Γ( )110

18.77. x e dxma

m axm

−+

∞= +∫ 2 1 2

2 1 20

Γ[( )/ ]( )/

18.78. e dxa

eax b x ab− +∞

−=∫ ( / )2 2 120

2π

18.79.x dx

ex − = + + + + =∞

∫ 111

12

13

14 60 2 2 2 2

2

�π

18.80.x

edx n

n

x n n n

−∞

− = + + +⎛⎝⎜

⎞⎠⎟∫

1

0 111

12

13

Γ( ) �

For even n this can be summed in terms of Bernoulli numbers (see pages 142–143).

18.81.x dx

ex + = − + − + =∞

∫ 111

12

13

14 120 2 2 2 2

2

�π

18.82.x

edx n

n

x n n n

−∞

+ = − + −⎛⎝⎜

⎞⎠⎟∫

1

0 111

12

13

Γ( ) �

For some positive integer values of n the series can be summed (see 23.10).

18.83.sin

cothmx

edx

mmx20 1

14 2

12π − = −

∞

∫

18.84.1

10 + −⎛⎝⎜

⎞⎠⎟ =−

∞

∫ xe

dxx

x γ

18.85.e e

xdx

x x− −∞ − =∫2

0

12 γ

18.86.1

10 eex

dxx

x

− −⎛⎝⎜

⎞⎠⎟

=−∞

∫ γ

18.87.e ex px

dxb pa p

ax bx− −∞ − = ++

⎛⎝⎜

⎞⎠⎟∫ sec

ln0

2 2

2 2

12

18.88.e ex px

dxbp

ap

ax bx− −∞− −− = −∫ csc

tan tan0

1 1

18.89. e xx

dx aa

aax−∞

−− = − +∫( cos )

cot ln ( )1

2120

1 2

DEFINITE INTEGRALS

114

Definite Integrals Involving Logarithmic Functions

18.90. x x dxn

mm nm n

n

n(ln )( ) !

( ), , , ,= −

+ > − =+∫1

11 0 1 210

1…

If n ≠ 0, 1, 2, … replace n! by Γ(n + 1).

18.91.ln x

xdx

1 12

2

0

1

+ = −∫π

18.92.ln x

xdx

1 6

2

0

1

− = −∫π

18.93.ln ( )1

120

1 2+ =∫x

xdx

π

18.94.ln ( )1

60

1 2− = −∫x

xdx

π

18.95. ln ln ( ) lnx x dx1 2 2 212

2

0

1+ = − −∫

π

18.96. ln ln ( )x x dx1 26

2

0

1− = −∫

π

18.97.x x

xdx p p p

p−∞

+ = − < <∫1

0

2

10 1

lncsc cotπ π π

18.98.x x

xdx

mn

m n− = ++∫ ln

ln0

1 11

18.99. e x dxx−∞

= −∫ ln γ0

18.100. e x dxx−∞

= − +∫ 2

42 2

0ln ( ln )

π γ

18.101. lnee

dxx

x

+−

⎛⎝⎜

⎞⎠⎟

=∞

∫11 40

2π

18.102. ln sin ln cos ln//

x dx x dx= = −∫∫πππ

22

0

2

0

2

18.103. (ln sin ) (ln cos ) (ln )/

x dx x dx2 2 23

0

2

0 22

24= = +∫

π ππππ /2

∫

18.104. x x dxln sin ln= −∫ππ 2

0 22

18.105. sin ln sin ln/

x x dx = −∫ 2 10

2π

18.106. ln ( sin ) ln ( cos ) ln ( )a b x dx a b x dx a a b+ = + = + −2 2 2

0

2π

πππ

∫∫0

2

18.107. ln( cos ) lna b x dxa a b+ = + −⎛

⎝⎜⎞

⎠⎟∫ ππ 2 2

0 2

DEFINITE INTEGRALS

115

18.108. ln ( cos )ln ,

ln ,a ab x b dx

a a b

b b a2 22

2 0

2 0− + =

>

>

⎧ π

π

�

�⎨⎨⎪

⎩⎪∫0

π

18.109. ln ( tan ) ln/

18

20

4+ =∫ x dx

ππ

18.110. sec lncoscos

{(cos ) (xb xa x

dx a11

12

1 2++

⎛⎝⎜

⎞⎠⎟

= −− ccos ) }/

−∫ 1 2

0

2b

π

18.111. ln sinsin sin sin

22 1

22

332 2 2

xdx

a a a⎛⎝⎜

⎞⎠⎟ = − + + +⎛ �⎝⎝⎜

⎞⎠⎟∫0

a

See also 18.102.

Definite Integrals Involving Hyperbolic Functions

18.112.sinsinh

tanhaxbx

dxb

ab

=∞

∫π π2 20

18.113.coscosh

axbx

dxb

ab

=∞

∫π π2 20

sech

18.114.x dx

ax asinh0

2

24∞

∫ = π

18.115.x dx

ax an

n n

n n n nsinh( )

0

1

1 1 1

2 12

11

11

2∞ +

+ + +∫ = − + +Γ ++ +{ }+1

3 1n �

If n is an odd positive integer, the series can be summed.

18.116.sinh

cscax

edx

bab abx + = −

∞

∫ 1 21

20

π π

18.117.sinh

cotax

edx

a babbx − = −

∞

∫ 11

2 20

π π

Miscellaneous Definite Integrals

18.118.f ax f bx

xdx f f

ba

( ) ( ){ ( ) ( )}ln

− = − ∞∞

∫00

This is called Frullani’s integral. It holds if f ′(x) is continuous and f x f

xdx

( ) ( )− ∞∞

∫0 converges.

18.119.dxx x = + + +∫

11

12

131 20

1

3 �

18.120. ( ) ( ) ( )( ) ( )( )

a x a x dx am nm n

m n m n+ − = +− − + −

−1 1 12

Γ ΓΓaa

a

∫

DEFINITE INTEGRALS

Tabla Integrales

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