Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture...
Transcript of Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture...
![Page 1: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/1.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where n is odd.
Strategy for integrating Zsinm x cosn xdx
We use substitution:
If n is odd (that is if the power of cosine is odd) we can use substitution withu = sin x , du = cos xdx and convert the remaining factors of cosine usingcos2 x = 1− sin2 x . This will work even if m = 0.Example Z
sin5 x cos3 xdx
I Let u = sin(x), du = cos x dx, cos2 x = 1− sin2 x = 1− u2.
IR
sin5 x cos3 xdx =R
sin5 x cos2 x cos x dx =R
(sin5 x)(1− sin2 x) cos x dx.
I =R
u5(1− u2) du =R
u5 − u7 du = u6
6− u8
8+ C = sin6 x
6− sin8 x
8+ C .
Annette Pilkington Trigonometric Integrals
![Page 2: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/2.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where n is odd.
Strategy for integrating Zsinm x cosn xdx
We use substitution:
If n is odd (that is if the power of cosine is odd) we can use substitution withu = sin x , du = cos xdx and convert the remaining factors of cosine usingcos2 x = 1− sin2 x . This will work even if m = 0.Example Z
sin5 x cos3 xdx
I Let u = sin(x), du = cos x dx, cos2 x = 1− sin2 x = 1− u2.
IR
sin5 x cos3 xdx =R
sin5 x cos2 x cos x dx =R
(sin5 x)(1− sin2 x) cos x dx.
I =R
u5(1− u2) du =R
u5 − u7 du = u6
6− u8
8+ C = sin6 x
6− sin8 x
8+ C .
Annette Pilkington Trigonometric Integrals
![Page 3: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/3.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where n is odd.
Strategy for integrating Zsinm x cosn xdx
We use substitution:
If n is odd (that is if the power of cosine is odd) we can use substitution withu = sin x , du = cos xdx and convert the remaining factors of cosine usingcos2 x = 1− sin2 x . This will work even if m = 0.Example Z
sin5 x cos3 xdx
I Let u = sin(x), du = cos x dx, cos2 x = 1− sin2 x = 1− u2.
IR
sin5 x cos3 xdx =R
sin5 x cos2 x cos x dx =R
(sin5 x)(1− sin2 x) cos x dx.
I =R
u5(1− u2) du =R
u5 − u7 du = u6
6− u8
8+ C = sin6 x
6− sin8 x
8+ C .
Annette Pilkington Trigonometric Integrals
![Page 4: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/4.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where n is odd.
Strategy for integrating Zsinm x cosn xdx
We use substitution:
If n is odd (that is if the power of cosine is odd) we can use substitution withu = sin x , du = cos xdx and convert the remaining factors of cosine usingcos2 x = 1− sin2 x . This will work even if m = 0.Example Z
sin5 x cos3 xdx
I Let u = sin(x), du = cos x dx, cos2 x = 1− sin2 x = 1− u2.
IR
sin5 x cos3 xdx =R
sin5 x cos2 x cos x dx =R
(sin5 x)(1− sin2 x) cos x dx.
I =R
u5(1− u2) du =R
u5 − u7 du = u6
6− u8
8+ C = sin6 x
6− sin8 x
8+ C .
Annette Pilkington Trigonometric Integrals
![Page 5: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/5.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where n is odd.
The substitution u = sin x works even if m = 0 and we have an odd power ofcosine.Example
Rcos5 x dx .
IR
cos5 x dx =R
cos4 x cos x dx =R
(cos2 x)2 cos x dx =R(1− sin2 x)2 cos x dx .
I Let u = sin x, du = cos x dx.
I ThenRcos5 x dx =
R(1− sin2 x)2 cos x dx =
R(1− u2)2 du =
R1− 2u2 + u4 du
= u − 2u3
3+
u5
5+ C = sin x − 2
sin3 x
3+
sin5 x
5+ C .
Annette Pilkington Trigonometric Integrals
![Page 6: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/6.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where n is odd.
The substitution u = sin x works even if m = 0 and we have an odd power ofcosine.Example
Rcos5 x dx .
IR
cos5 x dx =R
cos4 x cos x dx =R
(cos2 x)2 cos x dx =R(1− sin2 x)2 cos x dx .
I Let u = sin x, du = cos x dx.
I ThenRcos5 x dx =
R(1− sin2 x)2 cos x dx =
R(1− u2)2 du =
R1− 2u2 + u4 du
= u − 2u3
3+
u5
5+ C = sin x − 2
sin3 x
3+
sin5 x
5+ C .
Annette Pilkington Trigonometric Integrals
![Page 7: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/7.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where n is odd.
The substitution u = sin x works even if m = 0 and we have an odd power ofcosine.Example
Rcos5 x dx .
IR
cos5 x dx =R
cos4 x cos x dx =R
(cos2 x)2 cos x dx =R(1− sin2 x)2 cos x dx .
I Let u = sin x, du = cos x dx.
I ThenRcos5 x dx =
R(1− sin2 x)2 cos x dx =
R(1− u2)2 du =
R1− 2u2 + u4 du
= u − 2u3
3+
u5
5+ C = sin x − 2
sin3 x
3+
sin5 x
5+ C .
Annette Pilkington Trigonometric Integrals
![Page 8: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/8.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where n is odd.
The substitution u = sin x works even if m = 0 and we have an odd power ofcosine.Example
Rcos5 x dx .
IR
cos5 x dx =R
cos4 x cos x dx =R
(cos2 x)2 cos x dx =R(1− sin2 x)2 cos x dx .
I Let u = sin x, du = cos x dx.
I ThenRcos5 x dx =
R(1− sin2 x)2 cos x dx =
R(1− u2)2 du =
R1− 2u2 + u4 du
= u − 2u3
3+
u5
5+ C = sin x − 2
sin3 x
3+
sin5 x
5+ C .
Annette Pilkington Trigonometric Integrals
![Page 9: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/9.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where m is odd.
Zsinm x cosn xdx
If m is odd (that is if the power of sine is odd) we can use substitution withu = cos x , du = − sin xdx and convert the remaining factors of sine usingsin2 x = 1− cos2 x . This will work even if n = 0.Example
Rsin5 x cos4 xdx . (Note that the power of cosine is even here, so the
substitution u = sin x will not work.)
I Let u = cos(x), du = − sin x dx, sin2 x = 1− cos2 x = 1− u2.
IR
sin5 x cos4 xdx =R
sin4 x cos4 x sin x dx =R
(sin2 x)2(cos4 x) sin x dx.
I =R
(1− cos2 x)2 cos4 x sin x dx =R
(1− u2)2u4(−1 du) =−R
(1− 2u2 + u4)u4 du
I = −R
u4 − 2u6 + u8 du = −h
u5
5− 2 u7
7+ u9
9
i+ C
I = −h
cos5 x5− 2 cos7 x
7+ cos9 x
9
i+ C .
I Note the substitution u = cos x will work for odd powers of the sinefunction. See for example
Rsin3 x dx in the extra examples at the end of
your notes.
Annette Pilkington Trigonometric Integrals
![Page 10: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/10.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where m is odd.
Zsinm x cosn xdx
If m is odd (that is if the power of sine is odd) we can use substitution withu = cos x , du = − sin xdx and convert the remaining factors of sine usingsin2 x = 1− cos2 x . This will work even if n = 0.Example
Rsin5 x cos4 xdx . (Note that the power of cosine is even here, so the
substitution u = sin x will not work.)
I Let u = cos(x), du = − sin x dx, sin2 x = 1− cos2 x = 1− u2.
IR
sin5 x cos4 xdx =R
sin4 x cos4 x sin x dx =R
(sin2 x)2(cos4 x) sin x dx.
I =R
(1− cos2 x)2 cos4 x sin x dx =R
(1− u2)2u4(−1 du) =−R
(1− 2u2 + u4)u4 du
I = −R
u4 − 2u6 + u8 du = −h
u5
5− 2 u7
7+ u9
9
i+ C
I = −h
cos5 x5− 2 cos7 x
7+ cos9 x
9
i+ C .
I Note the substitution u = cos x will work for odd powers of the sinefunction. See for example
Rsin3 x dx in the extra examples at the end of
your notes.
Annette Pilkington Trigonometric Integrals
![Page 11: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/11.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where m is odd.
Zsinm x cosn xdx
If m is odd (that is if the power of sine is odd) we can use substitution withu = cos x , du = − sin xdx and convert the remaining factors of sine usingsin2 x = 1− cos2 x . This will work even if n = 0.Example
Rsin5 x cos4 xdx . (Note that the power of cosine is even here, so the
substitution u = sin x will not work.)
I Let u = cos(x), du = − sin x dx, sin2 x = 1− cos2 x = 1− u2.
IR
sin5 x cos4 xdx =R
sin4 x cos4 x sin x dx =R
(sin2 x)2(cos4 x) sin x dx.
I =R
(1− cos2 x)2 cos4 x sin x dx =R
(1− u2)2u4(−1 du) =−R
(1− 2u2 + u4)u4 du
I = −R
u4 − 2u6 + u8 du = −h
u5
5− 2 u7
7+ u9
9
i+ C
I = −h
cos5 x5− 2 cos7 x
7+ cos9 x
9
i+ C .
I Note the substitution u = cos x will work for odd powers of the sinefunction. See for example
Rsin3 x dx in the extra examples at the end of
your notes.
Annette Pilkington Trigonometric Integrals
![Page 12: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/12.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where m is odd.
Zsinm x cosn xdx
If m is odd (that is if the power of sine is odd) we can use substitution withu = cos x , du = − sin xdx and convert the remaining factors of sine usingsin2 x = 1− cos2 x . This will work even if n = 0.Example
Rsin5 x cos4 xdx . (Note that the power of cosine is even here, so the
substitution u = sin x will not work.)
I Let u = cos(x), du = − sin x dx, sin2 x = 1− cos2 x = 1− u2.
IR
sin5 x cos4 xdx =R
sin4 x cos4 x sin x dx =R
(sin2 x)2(cos4 x) sin x dx.
I =R
(1− cos2 x)2 cos4 x sin x dx =R
(1− u2)2u4(−1 du) =−R
(1− 2u2 + u4)u4 du
I = −R
u4 − 2u6 + u8 du = −h
u5
5− 2 u7
7+ u9
9
i+ C
I = −h
cos5 x5− 2 cos7 x
7+ cos9 x
9
i+ C .
I Note the substitution u = cos x will work for odd powers of the sinefunction. See for example
Rsin3 x dx in the extra examples at the end of
your notes.
Annette Pilkington Trigonometric Integrals
![Page 13: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/13.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where m is odd.
Zsinm x cosn xdx
If m is odd (that is if the power of sine is odd) we can use substitution withu = cos x , du = − sin xdx and convert the remaining factors of sine usingsin2 x = 1− cos2 x . This will work even if n = 0.Example
Rsin5 x cos4 xdx . (Note that the power of cosine is even here, so the
substitution u = sin x will not work.)
I Let u = cos(x), du = − sin x dx, sin2 x = 1− cos2 x = 1− u2.
IR
sin5 x cos4 xdx =R
sin4 x cos4 x sin x dx =R
(sin2 x)2(cos4 x) sin x dx.
I =R
(1− cos2 x)2 cos4 x sin x dx =R
(1− u2)2u4(−1 du) =−R
(1− 2u2 + u4)u4 du
I = −R
u4 − 2u6 + u8 du = −h
u5
5− 2 u7
7+ u9
9
i+ C
I = −h
cos5 x5− 2 cos7 x
7+ cos9 x
9
i+ C .
I Note the substitution u = cos x will work for odd powers of the sinefunction. See for example
Rsin3 x dx in the extra examples at the end of
your notes.
Annette Pilkington Trigonometric Integrals
![Page 14: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/14.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where m is odd.
Zsinm x cosn xdx
If m is odd (that is if the power of sine is odd) we can use substitution withu = cos x , du = − sin xdx and convert the remaining factors of sine usingsin2 x = 1− cos2 x . This will work even if n = 0.Example
Rsin5 x cos4 xdx . (Note that the power of cosine is even here, so the
substitution u = sin x will not work.)
I Let u = cos(x), du = − sin x dx, sin2 x = 1− cos2 x = 1− u2.
IR
sin5 x cos4 xdx =R
sin4 x cos4 x sin x dx =R
(sin2 x)2(cos4 x) sin x dx.
I =R
(1− cos2 x)2 cos4 x sin x dx =R
(1− u2)2u4(−1 du) =−R
(1− 2u2 + u4)u4 du
I = −R
u4 − 2u6 + u8 du = −h
u5
5− 2 u7
7+ u9
9
i+ C
I = −h
cos5 x5− 2 cos7 x
7+ cos9 x
9
i+ C .
I Note the substitution u = cos x will work for odd powers of the sinefunction. See for example
Rsin3 x dx in the extra examples at the end of
your notes.
Annette Pilkington Trigonometric Integrals
![Page 15: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/15.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where m is odd.
Zsinm x cosn xdx
If m is odd (that is if the power of sine is odd) we can use substitution withu = cos x , du = − sin xdx and convert the remaining factors of sine usingsin2 x = 1− cos2 x . This will work even if n = 0.Example
Rsin5 x cos4 xdx . (Note that the power of cosine is even here, so the
substitution u = sin x will not work.)
I Let u = cos(x), du = − sin x dx, sin2 x = 1− cos2 x = 1− u2.
IR
sin5 x cos4 xdx =R
sin4 x cos4 x sin x dx =R
(sin2 x)2(cos4 x) sin x dx.
I =R
(1− cos2 x)2 cos4 x sin x dx =R
(1− u2)2u4(−1 du) =−R
(1− 2u2 + u4)u4 du
I = −R
u4 − 2u6 + u8 du = −h
u5
5− 2 u7
7+ u9
9
i+ C
I = −h
cos5 x5− 2 cos7 x
7+ cos9 x
9
i+ C .
I Note the substitution u = cos x will work for odd powers of the sinefunction. See for example
Rsin3 x dx in the extra examples at the end of
your notes.
Annette Pilkington Trigonometric Integrals
![Page 16: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/16.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where both m and n are even.
Zsinm x cosn xdx
If both powers are even we reduce the powers using the half angle formulas:
sin2 x =1
2(1− cos 2x) cos2 x =
1
2(1 + cos 2x)
Alternatively, you can switch to powers of sine and cosine usingcos2 x + sin2 x = 1 and use the reduction formulas from the previous section.Example
Rsin2 x cos2 xdx .
IR
sin2 x cos2 xdx =R
[ 12(1− cos 2x)][ 1
2(1 + cos 2x)] dx =
14
R[1− cos2(2x)] dx
I Now we can use the half angle formula again: cos2(2x) = 12(1 + cos 4x).
IR
sin2 x cos2 xdx = 14
R “1− cos2(2x)
”dx = 1
4
R “1− [ 1
2(1 + cos 4x)]
”dx
I = 14
R12− 1
2cos 4x dx = 1
8
R1− cos 4x dx = x
8− sin 4x
32+ C
I See also the examplesR
sin4 x cos2 x dx andR
sin2 x dx in the extraproblems at the end of your notes.
Annette Pilkington Trigonometric Integrals
![Page 17: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/17.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where both m and n are even.
Zsinm x cosn xdx
If both powers are even we reduce the powers using the half angle formulas:
sin2 x =1
2(1− cos 2x) cos2 x =
1
2(1 + cos 2x)
Alternatively, you can switch to powers of sine and cosine usingcos2 x + sin2 x = 1 and use the reduction formulas from the previous section.Example
Rsin2 x cos2 xdx .
IR
sin2 x cos2 xdx =R
[ 12(1− cos 2x)][ 1
2(1 + cos 2x)] dx =
14
R[1− cos2(2x)] dx
I Now we can use the half angle formula again: cos2(2x) = 12(1 + cos 4x).
IR
sin2 x cos2 xdx = 14
R “1− cos2(2x)
”dx = 1
4
R “1− [ 1
2(1 + cos 4x)]
”dx
I = 14
R12− 1
2cos 4x dx = 1
8
R1− cos 4x dx = x
8− sin 4x
32+ C
I See also the examplesR
sin4 x cos2 x dx andR
sin2 x dx in the extraproblems at the end of your notes.
Annette Pilkington Trigonometric Integrals
![Page 18: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/18.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where both m and n are even.
Zsinm x cosn xdx
If both powers are even we reduce the powers using the half angle formulas:
sin2 x =1
2(1− cos 2x) cos2 x =
1
2(1 + cos 2x)
Alternatively, you can switch to powers of sine and cosine usingcos2 x + sin2 x = 1 and use the reduction formulas from the previous section.Example
Rsin2 x cos2 xdx .
IR
sin2 x cos2 xdx =R
[ 12(1− cos 2x)][ 1
2(1 + cos 2x)] dx =
14
R[1− cos2(2x)] dx
I Now we can use the half angle formula again: cos2(2x) = 12(1 + cos 4x).
IR
sin2 x cos2 xdx = 14
R “1− cos2(2x)
”dx = 1
4
R “1− [ 1
2(1 + cos 4x)]
”dx
I = 14
R12− 1
2cos 4x dx = 1
8
R1− cos 4x dx = x
8− sin 4x
32+ C
I See also the examplesR
sin4 x cos2 x dx andR
sin2 x dx in the extraproblems at the end of your notes.
Annette Pilkington Trigonometric Integrals
![Page 19: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/19.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where both m and n are even.
Zsinm x cosn xdx
If both powers are even we reduce the powers using the half angle formulas:
sin2 x =1
2(1− cos 2x) cos2 x =
1
2(1 + cos 2x)
Alternatively, you can switch to powers of sine and cosine usingcos2 x + sin2 x = 1 and use the reduction formulas from the previous section.Example
Rsin2 x cos2 xdx .
IR
sin2 x cos2 xdx =R
[ 12(1− cos 2x)][ 1
2(1 + cos 2x)] dx =
14
R[1− cos2(2x)] dx
I Now we can use the half angle formula again: cos2(2x) = 12(1 + cos 4x).
IR
sin2 x cos2 xdx = 14
R “1− cos2(2x)
”dx = 1
4
R “1− [ 1
2(1 + cos 4x)]
”dx
I = 14
R12− 1
2cos 4x dx = 1
8
R1− cos 4x dx = x
8− sin 4x
32+ C
I See also the examplesR
sin4 x cos2 x dx andR
sin2 x dx in the extraproblems at the end of your notes.
Annette Pilkington Trigonometric Integrals
![Page 20: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/20.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where both m and n are even.
Zsinm x cosn xdx
If both powers are even we reduce the powers using the half angle formulas:
sin2 x =1
2(1− cos 2x) cos2 x =
1
2(1 + cos 2x)
Alternatively, you can switch to powers of sine and cosine usingcos2 x + sin2 x = 1 and use the reduction formulas from the previous section.Example
Rsin2 x cos2 xdx .
IR
sin2 x cos2 xdx =R
[ 12(1− cos 2x)][ 1
2(1 + cos 2x)] dx =
14
R[1− cos2(2x)] dx
I Now we can use the half angle formula again: cos2(2x) = 12(1 + cos 4x).
IR
sin2 x cos2 xdx = 14
R “1− cos2(2x)
”dx = 1
4
R “1− [ 1
2(1 + cos 4x)]
”dx
I = 14
R12− 1
2cos 4x dx = 1
8
R1− cos 4x dx = x
8− sin 4x
32+ C
I See also the examplesR
sin4 x cos2 x dx andR
sin2 x dx in the extraproblems at the end of your notes.
Annette Pilkington Trigonometric Integrals
![Page 21: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/21.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where both m and n are even.
Zsinm x cosn xdx
If both powers are even we reduce the powers using the half angle formulas:
sin2 x =1
2(1− cos 2x) cos2 x =
1
2(1 + cos 2x)
Alternatively, you can switch to powers of sine and cosine usingcos2 x + sin2 x = 1 and use the reduction formulas from the previous section.Example
Rsin2 x cos2 xdx .
IR
sin2 x cos2 xdx =R
[ 12(1− cos 2x)][ 1
2(1 + cos 2x)] dx =
14
R[1− cos2(2x)] dx
I Now we can use the half angle formula again: cos2(2x) = 12(1 + cos 4x).
IR
sin2 x cos2 xdx = 14
R “1− cos2(2x)
”dx = 1
4
R “1− [ 1
2(1 + cos 4x)]
”dx
I = 14
R12− 1
2cos 4x dx = 1
8
R1− cos 4x dx = x
8− sin 4x
32+ C
I See also the examplesR
sin4 x cos2 x dx andR
sin2 x dx in the extraproblems at the end of your notes.
Annette Pilkington Trigonometric Integrals
![Page 22: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/22.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where both m and n are even.
Zsinm x cosn xdx
Note If both powers are even, as an alternative to using the half angleformulas, you can switch to powers of sine and cosine using cos2 x + sin2 x = 1and use the reduction formulas which can be derived using integration by parts:Z
cosn x =1
n[cosn−1 x sin x + (n − 1)
Zcosn−2 xdx ]
Zsinn xdx =
−1
n[sinn−1 x cos x − (n − 1)
Zsinn−2 xdx ]
ExampleR
sin2 x cos2 xdx .
IR
sin2 x cos2 xdx =R
[1− cos2 x ][cos2 x ] dx =R
[cos2 x − cos4(x)] dx
I We can then integrate cos2 x using the half angle formula and reduce theintegral of cos4 x to that of cos2 x using the reduction formula above.
Annette Pilkington Trigonometric Integrals
![Page 23: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/23.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where both m and n are even.
Zsinm x cosn xdx
Note If both powers are even, as an alternative to using the half angleformulas, you can switch to powers of sine and cosine using cos2 x + sin2 x = 1and use the reduction formulas which can be derived using integration by parts:Z
cosn x =1
n[cosn−1 x sin x + (n − 1)
Zcosn−2 xdx ]
Zsinn xdx =
−1
n[sinn−1 x cos x − (n − 1)
Zsinn−2 xdx ]
ExampleR
sin2 x cos2 xdx .
IR
sin2 x cos2 xdx =R
[1− cos2 x ][cos2 x ] dx =R
[cos2 x − cos4(x)] dx
I We can then integrate cos2 x using the half angle formula and reduce theintegral of cos4 x to that of cos2 x using the reduction formula above.
Annette Pilkington Trigonometric Integrals
![Page 24: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/24.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sinm x cosn xdx , where both m and n are even.
Zsinm x cosn xdx
Note If both powers are even, as an alternative to using the half angleformulas, you can switch to powers of sine and cosine using cos2 x + sin2 x = 1and use the reduction formulas which can be derived using integration by parts:Z
cosn x =1
n[cosn−1 x sin x + (n − 1)
Zcosn−2 xdx ]
Zsinn xdx =
−1
n[sinn−1 x cos x − (n − 1)
Zsinn−2 xdx ]
ExampleR
sin2 x cos2 xdx .
IR
sin2 x cos2 xdx =R
[1− cos2 x ][cos2 x ] dx =R
[cos2 x − cos4(x)] dx
I We can then integrate cos2 x using the half angle formula and reduce theintegral of cos4 x to that of cos2 x using the reduction formula above.
Annette Pilkington Trigonometric Integrals
![Page 25: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/25.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If m is even and m > 0, use substitution with u = tan x , and use one factor ofsec2 x for du = sec2 dx . Use sec2 x = 1 + tan2 x to convert the remainingfactors of sec2 x to a function of u = tan x . This works even if n = 0 as long asm ≥ 4.
ExampleR
sec4 x tan xdx
IR
sec4 x tan xdx =R
sec2 x sec2 x tan x dx.
I Let u = tan x, du = sec2 x dx, sec2 x = 1 + tan2 x.
IR
sec2 x sec2 x tan x dx =R
[1 + tan2 x ] tan x sec2 x dx =R
[1 + u2]u du
I =R
[u + u3] du = u2
2+ u4
4+ C = tan2 x
2+ tan4 x
4+ C .
Annette Pilkington Trigonometric Integrals
![Page 26: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/26.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If m is even and m > 0, use substitution with u = tan x , and use one factor ofsec2 x for du = sec2 dx . Use sec2 x = 1 + tan2 x to convert the remainingfactors of sec2 x to a function of u = tan x . This works even if n = 0 as long asm ≥ 4.
ExampleR
sec4 x tan xdx
IR
sec4 x tan xdx =R
sec2 x sec2 x tan x dx.
I Let u = tan x, du = sec2 x dx, sec2 x = 1 + tan2 x.
IR
sec2 x sec2 x tan x dx =R
[1 + tan2 x ] tan x sec2 x dx =R
[1 + u2]u du
I =R
[u + u3] du = u2
2+ u4
4+ C = tan2 x
2+ tan4 x
4+ C .
Annette Pilkington Trigonometric Integrals
![Page 27: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/27.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If m is even and m > 0, use substitution with u = tan x , and use one factor ofsec2 x for du = sec2 dx . Use sec2 x = 1 + tan2 x to convert the remainingfactors of sec2 x to a function of u = tan x . This works even if n = 0 as long asm ≥ 4.
ExampleR
sec4 x tan xdx
IR
sec4 x tan xdx =R
sec2 x sec2 x tan x dx.
I Let u = tan x, du = sec2 x dx, sec2 x = 1 + tan2 x.
IR
sec2 x sec2 x tan x dx =R
[1 + tan2 x ] tan x sec2 x dx =R
[1 + u2]u du
I =R
[u + u3] du = u2
2+ u4
4+ C = tan2 x
2+ tan4 x
4+ C .
Annette Pilkington Trigonometric Integrals
![Page 28: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/28.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If m is even and m > 0, use substitution with u = tan x , and use one factor ofsec2 x for du = sec2 dx . Use sec2 x = 1 + tan2 x to convert the remainingfactors of sec2 x to a function of u = tan x . This works even if n = 0 as long asm ≥ 4.
ExampleR
sec4 x tan xdx
IR
sec4 x tan xdx =R
sec2 x sec2 x tan x dx.
I Let u = tan x, du = sec2 x dx, sec2 x = 1 + tan2 x.
IR
sec2 x sec2 x tan x dx =R
[1 + tan2 x ] tan x sec2 x dx =R
[1 + u2]u du
I =R
[u + u3] du = u2
2+ u4
4+ C = tan2 x
2+ tan4 x
4+ C .
Annette Pilkington Trigonometric Integrals
![Page 29: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/29.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If m is even and m > 0, use substitution with u = tan x , and use one factor ofsec2 x for du = sec2 dx . Use sec2 x = 1 + tan2 x to convert the remainingfactors of sec2 x to a function of u = tan x . This works even if n = 0 as long asm ≥ 4.
ExampleR
sec4 x tan xdx
IR
sec4 x tan xdx =R
sec2 x sec2 x tan x dx.
I Let u = tan x, du = sec2 x dx, sec2 x = 1 + tan2 x.
IR
sec2 x sec2 x tan x dx =R
[1 + tan2 x ] tan x sec2 x dx =R
[1 + u2]u du
I =R
[u + u3] du = u2
2+ u4
4+ C = tan2 x
2+ tan4 x
4+ C .
Annette Pilkington Trigonometric Integrals
![Page 30: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/30.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If n is odd and m ≥ 1 use substitution with u = sec x , du = sec x tan x , andconvert remaining powers of tan to a function of u using tan2 x = sec2 x − 1.This works as long as m ≥ 1.
ExampleR
sec3 x tan xdx .
IR
sec3 x tan xdx =R
sec2 x sec x tan x dx.
I Let u = sec x, du = sec x tan x dx.
IR
sec2 x sec x tan x dx =R
u2 du = u3
3+ C
I = sec3 x3
+ C .
I See alsoR
sec3 x tan5 x dx in the extra examples.
Annette Pilkington Trigonometric Integrals
![Page 31: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/31.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If n is odd and m ≥ 1 use substitution with u = sec x , du = sec x tan x , andconvert remaining powers of tan to a function of u using tan2 x = sec2 x − 1.This works as long as m ≥ 1.
ExampleR
sec3 x tan xdx .
IR
sec3 x tan xdx =R
sec2 x sec x tan x dx.
I Let u = sec x, du = sec x tan x dx.
IR
sec2 x sec x tan x dx =R
u2 du = u3
3+ C
I = sec3 x3
+ C .
I See alsoR
sec3 x tan5 x dx in the extra examples.
Annette Pilkington Trigonometric Integrals
![Page 32: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/32.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If n is odd and m ≥ 1 use substitution with u = sec x , du = sec x tan x , andconvert remaining powers of tan to a function of u using tan2 x = sec2 x − 1.This works as long as m ≥ 1.
ExampleR
sec3 x tan xdx .
IR
sec3 x tan xdx =R
sec2 x sec x tan x dx.
I Let u = sec x, du = sec x tan x dx.
IR
sec2 x sec x tan x dx =R
u2 du = u3
3+ C
I = sec3 x3
+ C .
I See alsoR
sec3 x tan5 x dx in the extra examples.
Annette Pilkington Trigonometric Integrals
![Page 33: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/33.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If n is odd and m ≥ 1 use substitution with u = sec x , du = sec x tan x , andconvert remaining powers of tan to a function of u using tan2 x = sec2 x − 1.This works as long as m ≥ 1.
ExampleR
sec3 x tan xdx .
IR
sec3 x tan xdx =R
sec2 x sec x tan x dx.
I Let u = sec x, du = sec x tan x dx.
IR
sec2 x sec x tan x dx =R
u2 du = u3
3+ C
I = sec3 x3
+ C .
I See alsoR
sec3 x tan5 x dx in the extra examples.
Annette Pilkington Trigonometric Integrals
![Page 34: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/34.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If n is odd and m ≥ 1 use substitution with u = sec x , du = sec x tan x , andconvert remaining powers of tan to a function of u using tan2 x = sec2 x − 1.This works as long as m ≥ 1.
ExampleR
sec3 x tan xdx .
IR
sec3 x tan xdx =R
sec2 x sec x tan x dx.
I Let u = sec x, du = sec x tan x dx.
IR
sec2 x sec x tan x dx =R
u2 du = u3
3+ C
I = sec3 x3
+ C .
I See alsoR
sec3 x tan5 x dx in the extra examples.
Annette Pilkington Trigonometric Integrals
![Page 35: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/35.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
boxed∫
secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If m odd and n is even we can reduce to powers of secant using the identitysec2 x = 1 + tan2 x .
ExampleR
sec x tan2 xdx (see integral of sec x and sec3 x below.)
IR
sec x tan2 xdx =R
sec x [sec2 x − 1] dx =R
sec3 x − sec x dx.
I You will see how to calculate these integrals in the ”powers of Secant”section below.
I See alsoR
sec3 x tan2 x dx in the extra examples.
Annette Pilkington Trigonometric Integrals
![Page 36: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/36.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
boxed∫
secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If m odd and n is even we can reduce to powers of secant using the identitysec2 x = 1 + tan2 x .
ExampleR
sec x tan2 xdx (see integral of sec x and sec3 x below.)
IR
sec x tan2 xdx =R
sec x [sec2 x − 1] dx =R
sec3 x − sec x dx.
I You will see how to calculate these integrals in the ”powers of Secant”section below.
I See alsoR
sec3 x tan2 x dx in the extra examples.
Annette Pilkington Trigonometric Integrals
![Page 37: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/37.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
boxed∫
secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If m odd and n is even we can reduce to powers of secant using the identitysec2 x = 1 + tan2 x .
ExampleR
sec x tan2 xdx (see integral of sec x and sec3 x below.)
IR
sec x tan2 xdx =R
sec x [sec2 x − 1] dx =R
sec3 x − sec x dx.
I You will see how to calculate these integrals in the ”powers of Secant”section below.
I See alsoR
sec3 x tan2 x dx in the extra examples.
Annette Pilkington Trigonometric Integrals
![Page 38: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/38.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
boxed∫
secm x tann xdx
Strategy for integrating Zsecm x tann xdx
If m odd and n is even we can reduce to powers of secant using the identitysec2 x = 1 + tan2 x .
ExampleR
sec x tan2 xdx (see integral of sec x and sec3 x below.)
IR
sec x tan2 xdx =R
sec x [sec2 x − 1] dx =R
sec3 x − sec x dx.
I You will see how to calculate these integrals in the ”powers of Secant”section below.
I See alsoR
sec3 x tan2 x dx in the extra examples.
Annette Pilkington Trigonometric Integrals
![Page 39: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/39.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
∫sin(mx) cos(nx)dx ,
∫sin(mx) sin(nx)dx ,
∫cos(mx) cos(nx)dx .
To evaluateZsin(mx) cos(nx)dx
Zsin(mx) sin(nx)dx
Zcos(mx) cos(nx)dx
we reverse the identities
sin((m − n)x) = sin(mx) cos(nx)− cos(mx) sin(nx)
sin((m + n)x) = sin mx cos nx + sin nx cos mx
cos((m − n)x) = cos(mx) cos(nx) + sin(nx) sin(mx)
cos((m + n)x) = cos(mx) cos(nx)− sin(nx) sin(mx)
to get
sin(mx) cos(nx) =1
2
ˆsin((m − n)x) + sin((m + n)x
˜sin(mx) sin(nx) =
1
2
ˆcos((m − n)x)− cos((m + n)x
˜cos(mx) cos(nx) =
1
2
ˆcos((m − n)x) + cos((m + n)x
˜Annette Pilkington Trigonometric Integrals
![Page 40: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/40.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sin(mx) cos(nx)dx .
ExampleR
sin 7x cos 3xdx
I We use sin(mx) cos(nx) = 12
ˆsin((m − n)x) + sin((m + n)x
˜.
IR
sin 7x cos 3xdx = 12
Rsin(4x) + sin(10x) dx
I = −12
hcos(4x)
4+ cos(10x)
10
i+ C.
I Also seeR
cos(8x) cos(2x) dx andR
sin(x) sin(2x) dx in the extraexamples.
Annette Pilkington Trigonometric Integrals
![Page 41: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/41.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sin(mx) cos(nx)dx .
ExampleR
sin 7x cos 3xdx
I We use sin(mx) cos(nx) = 12
ˆsin((m − n)x) + sin((m + n)x
˜.
IR
sin 7x cos 3xdx = 12
Rsin(4x) + sin(10x) dx
I = −12
hcos(4x)
4+ cos(10x)
10
i+ C.
I Also seeR
cos(8x) cos(2x) dx andR
sin(x) sin(2x) dx in the extraexamples.
Annette Pilkington Trigonometric Integrals
![Page 42: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/42.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sin(mx) cos(nx)dx .
ExampleR
sin 7x cos 3xdx
I We use sin(mx) cos(nx) = 12
ˆsin((m − n)x) + sin((m + n)x
˜.
IR
sin 7x cos 3xdx = 12
Rsin(4x) + sin(10x) dx
I = −12
hcos(4x)
4+ cos(10x)
10
i+ C.
I Also seeR
cos(8x) cos(2x) dx andR
sin(x) sin(2x) dx in the extraexamples.
Annette Pilkington Trigonometric Integrals
![Page 43: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/43.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sin(mx) cos(nx)dx .
ExampleR
sin 7x cos 3xdx
I We use sin(mx) cos(nx) = 12
ˆsin((m − n)x) + sin((m + n)x
˜.
IR
sin 7x cos 3xdx = 12
Rsin(4x) + sin(10x) dx
I = −12
hcos(4x)
4+ cos(10x)
10
i+ C.
I Also seeR
cos(8x) cos(2x) dx andR
sin(x) sin(2x) dx in the extraexamples.
Annette Pilkington Trigonometric Integrals
![Page 44: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/44.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent∫sin(mx) cos(nx)dx .
ExampleR
sin 7x cos 3xdx
I We use sin(mx) cos(nx) = 12
ˆsin((m − n)x) + sin((m + n)x
˜.
IR
sin 7x cos 3xdx = 12
Rsin(4x) + sin(10x) dx
I = −12
hcos(4x)
4+ cos(10x)
10
i+ C.
I Also seeR
cos(8x) cos(2x) dx andR
sin(x) sin(2x) dx in the extraexamples.
Annette Pilkington Trigonometric Integrals
![Page 45: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/45.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Secant
Example Zsec xdx = ln | sec x + tan x |+ C
IR
sec xdx =R
sec x“
sec x+tan xsec x+tan x
”dx =
Rsec2 x+sec x tan x
sec x+tan xdx
I Using the substitution u = sec x + tan x, we get du = sec2 x + sec x tan xgiving us that the above integral isZ
1
udu = ln |u| = ln | sec x + tan x |+ C .
Zsec2 x dx = tan x + C .
Annette Pilkington Trigonometric Integrals
![Page 46: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/46.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Secant
Example Zsec xdx = ln | sec x + tan x |+ C
IR
sec xdx =R
sec x“
sec x+tan xsec x+tan x
”dx =
Rsec2 x+sec x tan x
sec x+tan xdx
I Using the substitution u = sec x + tan x, we get du = sec2 x + sec x tan xgiving us that the above integral isZ
1
udu = ln |u| = ln | sec x + tan x |+ C .
Zsec2 x dx = tan x + C .
Annette Pilkington Trigonometric Integrals
![Page 47: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/47.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Secant
Example Zsec xdx = ln | sec x + tan x |+ C
IR
sec xdx =R
sec x“
sec x+tan xsec x+tan x
”dx =
Rsec2 x+sec x tan x
sec x+tan xdx
I Using the substitution u = sec x + tan x, we get du = sec2 x + sec x tan xgiving us that the above integral isZ
1
udu = ln |u| = ln | sec x + tan x |+ C .
Zsec2 x dx = tan x + C .
Annette Pilkington Trigonometric Integrals
![Page 48: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/48.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Secant
Example Zsec3 xdx
I We use integration by parts with u = sec x, dv = sec2 x dx. We getdu = sec x tan x dx and v = tan x.
IR
sec3 xdx =R
sec2 x sec x dx = sec x tan x −R
tan2 x sec x dx
I = sec x tan x−R
(sec2 x−1) sec x dx = sec x tan x−R
sec3 x dx+R
sec x dx
I Solving forR
sec3 x dx, we getZsec3 xdx =
sec x tan x
2+
1
2
Zsec1 xdx =
sec x tan x
2+
1
2ln | sec x+tan x |+C .
I In fact for n ≥ 3, we can derive a reduction formula for powers of sec inthis way: Z
secn xdx =secn−2 x tan x
n − 1+
n − 2
n − 1
Zsecn−2 xdx .
Annette Pilkington Trigonometric Integrals
![Page 49: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/49.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Secant
Example Zsec3 xdx
I We use integration by parts with u = sec x, dv = sec2 x dx. We getdu = sec x tan x dx and v = tan x.
IR
sec3 xdx =R
sec2 x sec x dx = sec x tan x −R
tan2 x sec x dx
I = sec x tan x−R
(sec2 x−1) sec x dx = sec x tan x−R
sec3 x dx+R
sec x dx
I Solving forR
sec3 x dx, we getZsec3 xdx =
sec x tan x
2+
1
2
Zsec1 xdx =
sec x tan x
2+
1
2ln | sec x+tan x |+C .
I In fact for n ≥ 3, we can derive a reduction formula for powers of sec inthis way: Z
secn xdx =secn−2 x tan x
n − 1+
n − 2
n − 1
Zsecn−2 xdx .
Annette Pilkington Trigonometric Integrals
![Page 50: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/50.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Secant
Example Zsec3 xdx
I We use integration by parts with u = sec x, dv = sec2 x dx. We getdu = sec x tan x dx and v = tan x.
IR
sec3 xdx =R
sec2 x sec x dx = sec x tan x −R
tan2 x sec x dx
I = sec x tan x−R
(sec2 x−1) sec x dx = sec x tan x−R
sec3 x dx+R
sec x dx
I Solving forR
sec3 x dx, we getZsec3 xdx =
sec x tan x
2+
1
2
Zsec1 xdx =
sec x tan x
2+
1
2ln | sec x+tan x |+C .
I In fact for n ≥ 3, we can derive a reduction formula for powers of sec inthis way: Z
secn xdx =secn−2 x tan x
n − 1+
n − 2
n − 1
Zsecn−2 xdx .
Annette Pilkington Trigonometric Integrals
![Page 51: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/51.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Secant
Example Zsec3 xdx
I We use integration by parts with u = sec x, dv = sec2 x dx. We getdu = sec x tan x dx and v = tan x.
IR
sec3 xdx =R
sec2 x sec x dx = sec x tan x −R
tan2 x sec x dx
I = sec x tan x−R
(sec2 x−1) sec x dx = sec x tan x−R
sec3 x dx+R
sec x dx
I Solving forR
sec3 x dx, we getZsec3 xdx =
sec x tan x
2+
1
2
Zsec1 xdx =
sec x tan x
2+
1
2ln | sec x+tan x |+C .
I In fact for n ≥ 3, we can derive a reduction formula for powers of sec inthis way: Z
secn xdx =secn−2 x tan x
n − 1+
n − 2
n − 1
Zsecn−2 xdx .
Annette Pilkington Trigonometric Integrals
![Page 52: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/52.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Secant
Example Zsec3 xdx
I We use integration by parts with u = sec x, dv = sec2 x dx. We getdu = sec x tan x dx and v = tan x.
IR
sec3 xdx =R
sec2 x sec x dx = sec x tan x −R
tan2 x sec x dx
I = sec x tan x−R
(sec2 x−1) sec x dx = sec x tan x−R
sec3 x dx+R
sec x dx
I Solving forR
sec3 x dx, we getZsec3 xdx =
sec x tan x
2+
1
2
Zsec1 xdx =
sec x tan x
2+
1
2ln | sec x+tan x |+C .
I In fact for n ≥ 3, we can derive a reduction formula for powers of sec inthis way: Z
secn xdx =secn−2 x tan x
n − 1+
n − 2
n − 1
Zsecn−2 xdx .
Annette Pilkington Trigonometric Integrals
![Page 53: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/53.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Secant
Example Zsec3 xdx
I We use integration by parts with u = sec x, dv = sec2 x dx. We getdu = sec x tan x dx and v = tan x.
IR
sec3 xdx =R
sec2 x sec x dx = sec x tan x −R
tan2 x sec x dx
I = sec x tan x−R
(sec2 x−1) sec x dx = sec x tan x−R
sec3 x dx+R
sec x dx
I Solving forR
sec3 x dx, we getZsec3 xdx =
sec x tan x
2+
1
2
Zsec1 xdx =
sec x tan x
2+
1
2ln | sec x+tan x |+C .
I In fact for n ≥ 3, we can derive a reduction formula for powers of sec inthis way: Z
secn xdx =secn−2 x tan x
n − 1+
n − 2
n − 1
Zsecn−2 xdx .
Annette Pilkington Trigonometric Integrals
![Page 54: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/54.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Tangent
Ztann xdx
IR
tan xdx=R
sin xcos x
dx
I Using the substitution u = cos x, we get du = − sin x we get
IR
sin xcos x
dx =R −1
udu = − ln |u| = ln | sec x |+ C .
IR
tan2 xdx=R
(sec2 x − 1)dx = tan x − x + C
IR
tan3 xdx=R
(sec2 x − 1) tan xdx =R
(sec2 x) tan xdx −R
tan xdx
= tan2 x2− ln | sec x |+ C = tan2 x
2+ ln | cos x |+ C .
I In fact for n ≥ 2, we can derive a reduction formula for powers of tan xusing this method:Z
tann xdx =tann−1 x
n − 1−Z
tann−2 xdx
Annette Pilkington Trigonometric Integrals
![Page 55: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/55.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Tangent
Ztann xdx
IR
tan xdx=R
sin xcos x
dx
I Using the substitution u = cos x, we get du = − sin x we get
IR
sin xcos x
dx =R −1
udu = − ln |u| = ln | sec x |+ C .
IR
tan2 xdx=R
(sec2 x − 1)dx = tan x − x + C
IR
tan3 xdx=R
(sec2 x − 1) tan xdx =R
(sec2 x) tan xdx −R
tan xdx
= tan2 x2− ln | sec x |+ C = tan2 x
2+ ln | cos x |+ C .
I In fact for n ≥ 2, we can derive a reduction formula for powers of tan xusing this method:Z
tann xdx =tann−1 x
n − 1−Z
tann−2 xdx
Annette Pilkington Trigonometric Integrals
![Page 56: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/56.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Tangent
Ztann xdx
IR
tan xdx=R
sin xcos x
dx
I Using the substitution u = cos x, we get du = − sin x we get
IR
sin xcos x
dx =R −1
udu = − ln |u| = ln | sec x |+ C .
IR
tan2 xdx=R
(sec2 x − 1)dx = tan x − x + C
IR
tan3 xdx=R
(sec2 x − 1) tan xdx =R
(sec2 x) tan xdx −R
tan xdx
= tan2 x2− ln | sec x |+ C = tan2 x
2+ ln | cos x |+ C .
I In fact for n ≥ 2, we can derive a reduction formula for powers of tan xusing this method:Z
tann xdx =tann−1 x
n − 1−Z
tann−2 xdx
Annette Pilkington Trigonometric Integrals
![Page 57: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/57.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Tangent
Ztann xdx
IR
tan xdx=R
sin xcos x
dx
I Using the substitution u = cos x, we get du = − sin x we get
IR
sin xcos x
dx =R −1
udu = − ln |u| = ln | sec x |+ C .
IR
tan2 xdx=R
(sec2 x − 1)dx = tan x − x + C
IR
tan3 xdx=R
(sec2 x − 1) tan xdx =R
(sec2 x) tan xdx −R
tan xdx
= tan2 x2− ln | sec x |+ C = tan2 x
2+ ln | cos x |+ C .
I In fact for n ≥ 2, we can derive a reduction formula for powers of tan xusing this method:Z
tann xdx =tann−1 x
n − 1−Z
tann−2 xdx
Annette Pilkington Trigonometric Integrals
![Page 58: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/58.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Tangent
Ztann xdx
IR
tan xdx=R
sin xcos x
dx
I Using the substitution u = cos x, we get du = − sin x we get
IR
sin xcos x
dx =R −1
udu = − ln |u| = ln | sec x |+ C .
IR
tan2 xdx=R
(sec2 x − 1)dx = tan x − x + C
IR
tan3 xdx=R
(sec2 x − 1) tan xdx =R
(sec2 x) tan xdx −R
tan xdx
= tan2 x2− ln | sec x |+ C = tan2 x
2+ ln | cos x |+ C .
I In fact for n ≥ 2, we can derive a reduction formula for powers of tan xusing this method:Z
tann xdx =tann−1 x
n − 1−Z
tann−2 xdx
Annette Pilkington Trigonometric Integrals
![Page 59: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/59.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Tangent
Ztann xdx
IR
tan xdx=R
sin xcos x
dx
I Using the substitution u = cos x, we get du = − sin x we get
IR
sin xcos x
dx =R −1
udu = − ln |u| = ln | sec x |+ C .
IR
tan2 xdx=R
(sec2 x − 1)dx = tan x − x + C
IR
tan3 xdx=R
(sec2 x − 1) tan xdx =R
(sec2 x) tan xdx −R
tan xdx
= tan2 x2− ln | sec x |+ C = tan2 x
2+ ln | cos x |+ C .
I In fact for n ≥ 2, we can derive a reduction formula for powers of tan xusing this method:Z
tann xdx =tann−1 x
n − 1−Z
tann−2 xdx
Annette Pilkington Trigonometric Integrals
![Page 60: Mixed Powers of secsin( ) sin( etcPowers of SecantPowers ...apilking/Math10560/Lectures/Lecture 9.pdf · Mixed powers ofR sin and cos Mixed Powers of tan and secsin(mx) sin(nx) etc....Powers](https://reader033.fdocuments.in/reader033/viewer/2022041422/5e1fbe5ca588f25f7b5a8cb7/html5/thumbnails/60.jpg)
Mixed powers of sin and cos Mixed Powers of tan and sec sin(mx) sin(nx) etc.... Powers of Secant Powers of Tangent
Powers of Tangent
Ztann xdx
IR
tan xdx=R
sin xcos x
dx
I Using the substitution u = cos x, we get du = − sin x we get
IR
sin xcos x
dx =R −1
udu = − ln |u| = ln | sec x |+ C .
IR
tan2 xdx=R
(sec2 x − 1)dx = tan x − x + C
IR
tan3 xdx=R
(sec2 x − 1) tan xdx =R
(sec2 x) tan xdx −R
tan xdx
= tan2 x2− ln | sec x |+ C = tan2 x
2+ ln | cos x |+ C .
I In fact for n ≥ 2, we can derive a reduction formula for powers of tan xusing this method:Z
tann xdx =tann−1 x
n − 1−Z
tann−2 xdx
Annette Pilkington Trigonometric Integrals