Calculus Compilation

7/26/2019 Calculus Compilation

1/21

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Calculus Compilation

My Finals Output

MICHELLE MENDOZA


2/21

1 Basic ntegration !ormulas

Sample 1.1

7dx

7dx=7x+C

Sample 1.2

x3 dx

x3dx=x4

4+C

Sample 1.

3x4+x29x+1dx

!e note t"e #unction an$ its anti$e%i&ati&e 'elo()

f(x )=3x4+x29x+1

F(x )=3x5

5+ x

3

39x

2

2+x+C

3x4+x29x+1dx=35

x5+

1

3x

39

2x

2+x+C

Sample 1.*

sinx dx

f(x )=sinx

F(x )=cosx+C

sinx dx=cosx+C


3/21

Sample 1.+

ex dx

ex dx=ex+C

Sample 1.,

e8x dx

e8x dx=1

8e8x+C

Sample 1.-

ex dx

ex dx= 11 ex+C=ex+C

Sample 1.

dxx

2+9

/"is is actually in t"e #o%m o# an a%ctan (it" t"e &alue o# a 'ein0 sua%e$

p%o$uces 34. 5ou 0et ans(e% o#)

dx

x

2

+9

=1

3tan

1(x3 )+C

Sample 1.3

(x+ 3x )dx


4/21

(x+ 3x ) dx=x dx+ 3x dx

x1

2 dx+x1

3 dx

x

1

2+1

1

2+1

+x

1

3+1

1

3+1

+C

2x

3

2

3 +

3x

4

3

4 +C

(x+3x ) dx=2x

3

3 +3

3

x4

4 +C

Sample 1.16

( 33x + 2

x)dx

(

3

3

x+ 2

x)dx= 3dx3

x+ 2dx

x

3x13 dx+2x

12 dx

3 x

13+1

13 +1

+2 x

12+1

12 +1

+C=9x

2

3

2 +4x

1

2+C

( 33x+ 2

x)dx=9

3

x2

2 +4x+C

" ntegration of #rigonometric !unction

Sample 2.1


5/21

sin2x dx

sin2x dx= 1cos2x2

dx

12

dx cos2x2

dx

1

2x

1

2 (1

2sin 2x )+C

x

2 sinx cos+C

1

2x

1

4

1

2x

1

2sinxcosx+C

Sample 2.2

e1+ sinxcosx dx

e1+ sinx cosx dx= eu du

eu+C

e1+sinx+C

Sample 2.

sin(2t)

t dt

sin(2t)t

dt= sin2u du


6/21

cos2u

2+C

cos(2t)

2+C

Sample 2.*

3u

2+sin4

cos3u

3u

2+sin4

cos3u

1

3 v4dv

1

9v3+C

1

9 v3+C


7/21

3u

2+sin39

1

Sample 2.+

cosaxcosbx dx

cosaxcosbx dx= cos(axbx )+cos(ax+bx )

2 dx

cos (ab )x+cos(a+b)(x )dx

1

2

1

2 (sin (ab )x

ab +

sin (a+b )xa+b )+C

Sample 2.,

3x

sec 4


8/21

3x

sec 3x

sec

2

3x

tan 3x

sec

2

(2)1+

Let u 7 tan 89 $u 7 sec84: $89 $8 7du

3 (sec3x)2

(1+u2 ) du= 13 (u+

1

3u

3

)+C1

3

3x

+C

1

3tan 3x+

1

9tan

Sample 2.-

sin6x cos3x dx


9/21

sin6x cos3xdx=sin6x cos2xcosx dx

sin6x(1sin2x)cosx dx

u6 (1u2 ) du

u6u8du

1

7sin

7x

1

9sin

9x+C

Sample 2.

cos (15x ) cos (4x ) dx

cos (15x ) cos (4x ) dx=12cos (11x )+cos (19x )dx

1

2 ( 1

11sin (11x )+

1

19sin (19x ))+C

note that weuse theformula:cos cos=12

[cos ( )+cos (+)]

Sample 2.3

tanx dx

tanx dx= sinxcosx

dx

1

udu

|cosx|+C


11/21

2x

3x

dx

(

2

3

)

x

dx

( 23 )

x

( 23 )+C

Sample .

cos

xsin

5

xdx

cosx sin5x d x= u5 du

1

6sin

6

x+C

Sample .*

3e4x

dx

3e4xdx=3 (eu) du4

3

4 eu du

3

4

eu+C

3

4e4x+C

Sample .+


12/21

ex4

4x3dx

Let u=x4,thendu=4x3dx

ex44x

3dx= e

u du

eu+C

ex

4

+C

Sample .,

xe

xdx

Let u=xv=ex

xex dx=uv v du

x exex+C

Sample .-

tan1x dx

Let u=tan1x , the du= 1

1+x2dxv=x

tan1x dx=uv v du

uv12 v du


13/21

x

tan1 (x )

1

2

Sample .

5x

xdx

5x

xdx=2 1

2x5

x dx

2

55

x

+C

Sample .3

etanx

cos2

xdx

etanx

cos2

xdx=e tanx+C

Sample .16

xex2

dx

xex2

dx=1

22xex

2

dx

1

2

ex2+C

& ntegration of 'ational !raction

Sample *.1


14/21

2x+3x

29dx

2x+3

(x3

)(x+3

)

3

2 dx

x3+1

2 dx

x+3

3

2|x3|+ 1

2|x+3|+C

1

2|(x3)3 (x+3 )|+C

Sample *.2

x22

x+1dx

x1 1x+1

xdxdx dxx+1

x

2

2x|x+1|+C

Sample *.

x2dx

(x1

)(x2

)(x3

)

1

2 dx

x14 dx

x2+9

2 dx

x3


15/21

1

2|x1|4|x2|+ 9

2|x3|+C

Sample *.+

5x(x1)3

dx

( 5(x1 )3+ 5

(x1)2 )dx

5 (x1)2

2

5

x1+C

5

2 (x1 )2

5

x1+C

Sample *.,

3+x2

x dx

(3

x+x

2

x)dx

3 1x

dx+xdx

3|x|+x2

2+C

Sample *.-

7x+5

dx

Let u=x+5


16/21

7udx

7|x+5|+C

Sample *.

1x(3+Inx)

dx

Let u=3+Inxdu=1

xdx

1

3+x

1

xdx

1udu

|3+Inx|+C

Sample *.3

7

Inxx (3+x )dx

Let u=3+Inxdu=1

xdx

7Inx3Inx

1

xdx

7

(u3

)u du

10uu

du=(10

u1)du


17/21

10|u|u+C

10|3+x|3x+C

Sample *.3

1x2

dx

Let u=x2du=dx

1udu=|u|+C

|x2|+C

Sample *.16

r22 r

r33 r2+1

Let u=r33r2+1 ,then du=3r 26r3 (r22r )

1

3 1

udu

1

3|r33r 2+1|+C

( )ummation *otations

Sample +.1

1

100

(32 i)2


18/21

1

100

912i+14 i2

1

100

912i(i+1)

2

+14i (i+1)(2 i+1)

6

1

100

96 i2+i+7

3(2 i3+3i2+i )

Sample +.2

k=1

n

3k2+5k+4

k=1

n

3(2n+1)n(n+1)

6+5

n (n+1)2

+4n

k=1

n2n

3+3n2+n2

+5n

2+n2

+4 n

Sample +.

3

7

4 i2

3

7

42 i

3+3 i2+i6

3

7

22 i

3+3 i2+i3

Sample +.*

0

3

1+i3


19/21

0

3

1+i2 (i+1 )2

4

0

3

1+i4+2i3+i2

4

Sample +.+

1

50

2k

1

50

2k(k+1)

2

1

50

k2+k

Sample +.,

k=1

n

6 k5

k=1

n

6k(k+1)2

5

k=1

n

3k2+3k5k

k=1

n

k(3 k2)

Sample +.-

k=1

n

4k3


20/21

k=1

n

4k(k+1)

23

k=1

n

2k2+2k3k

k=1

n

k(2k1)

Sample +.

x=1

n

3x2+4x5

x=1

n

3(2n+1)n(n+1)

6+4

n (n+1)2

5 n

x=1

n2n

3+3n2+n2

+2n23n

Sample +.3

i=5

3

2 i+1

i=5

3

2i(i+1)

2+1

i=5

3

i(i+2)

Sample +.16

1

n

a


21/21

1

n

an

,

Calculus Compilation

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