Higher Engineering Mathematics

400 INTEGRAL CALCULUS

=

[

3 θ

2

+sin 2θ+

sin 4θ

8

]π

4

0

=

[

3

2

(π

4

)

+sin

2 π

4

+

sin 4(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 +cos 2t

2

) 2

dt

=

1

8

∫

(1−cos 2t)(1+2 cos 2t+cos^22 t)dt

=

1

8

∫

(1+2 cos 2t+cos^22 t−cos 2t

−2 cos^22 t−cos^32 t)dt

=

1

8

∫

(1+cos 2t−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

−

sin 4t

8

+

sin^32 t

6

)

+c

Now try the following exercise.

Exercise 157 Further problems on integra-

tion of powers of sines and cosines

In Problems 1 to 6, integrate with respect to the

variable.

1. sin^3 θ

[

(a)−cosθ+

cos^3 θ

3

+c

]

2. 2 cos^32 x

[

sin 2x−

sin^32 x

3

+c

]

3. 2 sin^3 tcos^2 t [
   − 2
   3

cos^3 t+

2

5

cos^5 t+c

]

4. sin^3 xcos^4 x

[

−cos^5 x

5

+

cos^7 x

7

+c

]

5. 2 sin^42 θ [
   3 θ
   4

−

1

4

sin 4θ+

1

32

sin 8θ+c

]

6. sin^2 tcos^2 t

[

t

8

−

1

32

sin 4t+c

]

40.4 Worked problems on integration

of products of sines and cosines

Problem 9. Determine

∫

sin 3tcos 2tdt.

∫

sin 3tcos 2tdt

=

∫

1

2

[sin (3t+ 2 t)+sin (3t− 2 t)] dt,

from 6 of Table 40.1, which follows from Sec-

tion 18.4, page 183,

=

1

2

∫

(sin 5t+sint)dt

=

1

2

(

−cos 5t

5

−cost

)

+c

Problem 10. Find

∫

1

3

cos 5xsin 2xdx.

∫

1

3

cos 5xsin 2xdx

=

1

3

∫

1

2

[sin (5x+ 2 x)−sin (5x− 2 x)] dx,

from 7 of Table 40.1

=

1

6

∫

(sin 7x−sin 3x)dx

=

1

6

(

−cos 7x

7

+

cos 3x

3

)

+c