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: