CK-12-Calculus

http://www.ck12.org Chapter 7. Integration Techniques

∫

cos^2 xdx=

∫ 1

2 (^1 +cos 2x)dx

=^12

∫

( 1 +cos 2x)dx

=^12

(

x+^12 sin 2x

)

+C

=x 2 +^14 sin 2x+C.

Example 2:
Evaluate∫cos^4 xdx.
Solution:

∫

cos^4 xdx=

∫

(cos^2 x)^2 dx=

∫( 1

2 (^1 +cos 2x)

) 2

dx

=^14

∫

( 1 +2 cos 2x+cos^22 x)dx

=^14

∫(

1 +2 cos 2x+^12 +^12 cos 4x

)

dx

=^14

∫( 3

2 +2 cos 2x+

1

2 cos 4x

)

dx.

Integrating term by term,

=^14

[ 3

2 x+sin 2x+

1

8 sin 4x

]

+C

=^38 x+^14 sin 2x+ 321 sin 4x+C.

Example 3:
Evaluate∫sin^3 xdx.
Solution:

∫

sin^3 xdx=

∫

sin^2 xsinxdx

Recall that sin^2 x+cos^2 x= 1 ,so by substitution,

=

∫

( 1 −cos^2 x)sinxdx

=

∫

sinxdx−

∫

cos^2 xsinxdx.

The first integral should be straightforward. The second can be done by the method ofu−substitution by letting
u=cosx,sodu=−sinxdx.The integral becomes