http://www.ck12.org Chapter 7. Integration Techniques
Example 5:
Evaluate∫sin^4 xcos^4 xdx.
Solution:
Here,m=n= 4 .We follow the third procedure in the table above:
∫
sin^4 xcos^4 xdx=∫
(sin^2 x)^2 (cos^2 x)^2 dx=∫[ 1
2 (^1 −cos 2x)] 2 [ 1
2 (^1 +cos 2x)] 2
dx= 161∫
( 1 −cos^22 x)^2 dx
= 161∫
sin^42 xdx.At this stage, it is best to useu−substitution to integrate. Letu= 2 x,sodu= 2 dx.
∫
sin^4 xcos^4 xdx= 321∫
sin^4 udu= 321∫
(sin^2 u)^2 du= 321∫[ 1
2 (^1 −cos 2u)] 2
du= 321( 3
8 u−1
4 sin 2u+1
32 sin 4u)
+C
= 2563 x− 1281 sin 4x+ 10241 sin 8x+C.Integrating Powers of Secants and Tangents
In this section we will study methods of integrating functions of the form
∫
tanmxsecnxdx,wheremandnare nonnegative integers. However, we will begin with the integrals
∫
tanxdxand
∫
secxdx.The first integral can be evaluated by writing