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7.2 - Trigonometric Integrals
1. Consider integrals of the form:  sin n x cos m xdx
a. Either n or m is an odd positive integer:
n is an odd positive integer: n − 1 is even
 sin n x cos m xdx   sin n−1 x cos m x sin xdx
 − 1 − cos 2 x n−1/2 cos m xdcos x
m is an odd positive integer: m − 1 is even
 sin n x cos m xdx   sin n x cos m−1 x cos xdx

 sin n x1 − sin 2 x
m−1/2
dsin x
b. Both n and m are positive even numbers:
Use the identities:
sin 2 x  1 1 − cos2x, and cos 2 x  1 1  cos2x
2
2
to reduce the power of sin x and cos x.
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Example: (1)  cos 4 x sin 3 x dx (2)  sin 4 x cos 5 xdx
(1)  cos 4 x sin 3 x dx, u  cos x, du  − sin xdx
 cos 4 x sin 3 x dx  −  cos 4 x1 − cos 2 xdcos x  − u 4 − u 6 du
 − 1 u 5  1 u 7  C  − 1 cos 5 x  1 cos 7 x  C
5
7
5
7
(2)  sin 4 x cos 5 xdx   sin 4 x1 − sin 2 x dsin x, u  sin x, du  cos x
2
 sin 4 x cos 5 xdx   u 4 1 − 2u 2  u 4 du  u 4 − 2u 6  u 8 du
 1 u5 − 2 u7  1 u9  C
5
7
9
 1 sin 5 x − 2 sin 7 x  1 sin 9 x  C
5
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9
2
Example: Evaluate (1)  cos 2 x sin 2 xdx (2)  cos 2 2x sin 4 2xdx
(1)
 cos 2 x sin 2 xdx  
1 1  cos2x 1 1 − cos2xdx
2
2
 1 1 − cos 2 2xdx  1  1 − 1 1  cos4x dx
2
4
4
 1  1 − 1 cos4x dx  1 x − 1 sin4x  C
2
2
8
4
4
3
(2)




 cos 2 2x sin 4 2xdx   12 1  cos4x 14 1 − cos4x 2 dx
1 1  cos4x1 − 2 cos4x  cos 2 4xdx
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1 1  cos4x − 2 cos4x − 2 cos 2 4x  cos 2 4x − cos 3 4xdx
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1 1 − cos4x − cos 2 4x − cos 3 4xdx
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1  1 − cos4x − 1 1  cos8x dx −  1 1 − sin 2 4xd 1 sin4x
2
4
2
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 1
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1 x − 1 sin4x − 1 sin8x − 1 sin4x  1
2
4
16
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8
1 sin 3 4x
3
C
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2. Consider the integrals of the form:
 tan m x sec n xdx
a. n is an even positive numbers.
 tan m x sec n xdx   tan m x sec n−2 x sec 2 xdx
  tan m x1  tan 2  n−2/2 dtan x
b. m is an odd positive numbers.
 tan m x sec n xdx   tan m−1 x sec n−1 xtan x sec xdx
 sec 2 − 1 m−1/2 sec n−1 xdsec x
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Example: Evaluate (1)  tan 4 x sec 4 xdx
(2)  tan 5 x sec 2 xdx
(1) u  tan x, du  sec 2 xdx,
 tan 4 x sec 4 xdx

 tan 4 x1  tan 2 xdtan x   u 4 1  u 2 du
 1 u 5  1 u 7  C  1 tan 5 x  1 tan 7 x  C
5
7
5
7
(2) u  sec x, du  sec x tan xdx
 tan 5 x sec 2 xdx  sec 2 − 1 2 sec 2 xdsec x  u 4 − 2u 2  1u 2 du
 u 6 − 2u 4  u 2 du  1 u 7 − 2 u 5  1 u 3  C
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3
 1 sec 7 x − 2 sec 5 x  1 sec 3 x  C
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5
3
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Example: Evaluate  sec xdx
2 x  tan x sec x
sec
sec
x

tan
x
 sec xdx   sec x sec x  tan x dx   sec x  tan x dx
usec xtan x
1 du  ln|u |  C  ln|sec x  tan x |  C


u
dusec x tan xsec 2 xdx
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Example Evaluate  sec 3 xdx
 sec 3 xdx   secx sec 2 xdx
usecx,dvsec 2 xdx

dusecx tanxdx, vtanx
secx tanx −  secx tan 2 xdx
 secx tanx −  secxsec 2 x − 1dx
 secx tanx −  sec 3 xdx   secxdx
2  sec 3 xdx  secx tanx − ln|secx  tanx |  C
 sec 3 xdx 
1 secx tanx − ln|secx  tanx |   C
2
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3. Integrals of the forms:
(2)  sinax sinbxdx
(1)  sinax cosbxdx
 cosax cosbxdx
Identities:
(a) sin A cos B  12 sinA − B  sinA  B
(b) sin A sin B  12 cosA − B − cosA  B
(c) cos A cos B  12 cosA − B  cosA  B
(3)
Example: Evaluate  sin4x sin5xdx
 sin4x sin5xdx 
1 cos−x − cos9xdx
2
 1 sinx − 1 sin9x  C
2
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