http://www.ck12.org Chapter 7. Integration Techniques
Trigonometric Integrands
We can apply the change of variable technique to trigonometric functions as long asuis a differentiable function of
x.Before we show how, recall the basic trigonometric integrals:
∫
cosudu=sinu+C,
∫
sinudu=−cosu+C,
∫
sec^2 udu=tanu+C,
∫
csc^2 udu=−cotu+C,
∫
(secu)(tanu)du=secu+C,
∫
(cscu)(cotu)du=−cscu+C.Example 3:
Evaluate∫cos( 3 x+ 2 )dx.
Solution:
The argument of the cosine function is 3x+ 2 .So we letu= 3 x+ 2 .Thendu= 3 dx,ordx=du/ 3.
Substituting,
∫
cos( 3 x+ 2 )dx=∫
cosu·^13 dx
=^13∫
cosudx.Integrating,
=^13 sinu+C
=^13 sin( 3 x+ 2 )+C.Example 4:
This example requires us to use trigonometric identities before we substitute. Evaluate
∫ 1
cos^23 xdx.Solution:
Since sec 3x=cos 3^1 x, the integral becomes