Integration using trigonometric and hyperbolic substitutions 401
∫ π
4
0
4cos^4 θdθ= 4∫ π
4
0(cos^2 θ)^2 dθ= 4∫ π
4
0[
1
2( 1 +cos 2θ)] 2
dθ=∫ π
4
0( 1 +2cos2θ+cos^22 θ)dθ=∫ π
4
0[
1 +2cos2θ+1
2( 1 +cos 4θ)]
dθ=∫ π
4
0(
3
2+2cos2θ+1
2cos 4θ)
dθ=[
3 θ
2+sin2θ+sin4θ
8]π
4
0=[
3
2(π
4)
+sin2 π
4+sin4(π/ 4 )
8]
−[0]=3 π
8+ 1 = 2. 178 ,
correct to 4 significant figures.Problem 8. Find∫
sin^2 tcos^4 tdt.∫
sin^2 tcos^4 tdt=∫
sin^2 t(cos^2 t)^2 dt=∫ (
1 −cos 2t
2)(
1 +cos2t
2) 2
dt=1
8∫
( 1 −cos2t)( 1 +2cos2t+cos^22 t)dt=1
8∫
( 1 +2cos2t+cos^22 t−cos 2t
−2cos^22 t−cos^32 t)dt
=1
8∫
( 1 +cos2t−cos^22 t−cos^32 t)dt=1
8∫[
1 +cos 2t−(
1 +cos 4t
2)−cos 2t( 1 −sin^22 t)]
dt=1
8∫(
1
2−cos 4t
2+cos 2tsin^22 t)
dt=1
8(
t
2−sin4t
8+sin^32 t
6)
+cNow try the following exerciseExercise 156 Further problemson
integration of powersof sines and cosines
In Problems 1 to 6, integrate with respect to the
variable.- sin^3 θ
[
(a)−cosθ+cos^3 θ
3+c]- 2cos^32 x
[
sin2x−sin^32 x
3+c]- 2sin^3 tcos^2 t [
− 2
3
cos^3 t+2
5cos^5 t+c]- sin^3 xcos^4 x
[
−cos^5 x
5+cos^7 x
7+c]- 2sin^42 θ [
3 θ
4
−
1
4
sin4θ+1
32
sin8θ+c]- sin^2 tcos^2 t
[
t
8−1
32sin4t+c]40.4 Worked problems on integration
of products of sines and cosines
Problem 9. Determine∫
sin3tcos2tdt.
∫
sin3tcos 2tdt=∫
1
2[sin( 3 t+ 2 t)+sin( 3 t− 2 t)]dt,from 6 of Table 40.1, which follows from Section 17.4,
page 170,=1
2∫
(sin5t+sint)dt=1
2(
−cos 5t
5−cost)
+cProblem 10. Find∫
1
3
cos 5xsin2xdx.