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11
Example 5 Evaluate
Solution Since the exponents of both sin x and cos x are even, we use the
identities and to rewrite the integrand:
Use the identity to rewrite the integrand of the first integral.
Use the substitution u = sin 2x with du = 2cos 2x dx in the second integral.
. cos sin 42 dxxx) cos(sin x21
21
x2 = ) cos(cos x2121
x2 +=
. )cossin( cossin
coscos cos )cos cos)( cos(
) cos)( cos( ) cos() cos( cos sin
dxx2x2181
dxx281
x221
x81
dxx2x2x2181
dxx2x221x2181
dxx21x2181
dxx2121
x2121
dxxx
22
322
22
42
+=
+=++=
+=
+=
) cos(cos x4121
x22 +=
duu1161
x441
x161
x2161
8x
dxx22x2121
81
dxx4121
81
x221
x81
dxxx
2
242
sinsin
cos)sin( ) cos(sin cos sin
++=
+
+=
2
C3u
u161
x4641
x2161
16x
dxxx3
42 +
+= sinsin cos sin
duu1161
x441
x161
x2161
8x
dxxx 242 sinsin cos sin ++=
Cx2481
x4641
16x
Cx231
x2161
x4641
x2161
16x
dxxx
3
342
++=
+
+=sinsin
sinsinsinsin cos sin
Substitute u = sin 2x:
u = sin 2x: