Section 2.2 Calculus 2 - Professor Clontz

2.2 Integration of Trigonometric Products

To integrate a function of the form \(\sin^m x\cos^n x\) where sine’s power is odd, use the identity \(\sin^2 x=1-\cos^2 x\) to convert all but one sine to cosines. Then substitute \(u=\cos x,du=-\sin x\,dx\).
Example Find \(\int\sin^3\theta\cos^4\theta\,d\theta\).
If cosine’s power is odd, use the identity \(\cos^2 x=1-\sin^2 x\) to convert all but one cosine to sines. Then substitute \(u=\sin x,du=\cos x\,dx\).
Example Find \(\int\sin^2(2y)\cos^5(2y)\,dy\).

If both powers are even, then one or both of these identities must be used:
- \(\cos^2 x=\frac{1}{2}+\frac{1}{2}\cos(2x)\)
- \(\sin^2 x=\frac{1}{2}-\frac{1}{2}\cos(2x)\)
Example Find \(\int\cos^2 x\,dx\).
Example Find \(\int\sin^2 z\cos^2 z\,dz\).

To integrate a function of the form \(\sec^m x\tan^n x\) where tangent’s power is odd, use the identity \(\tan^2 x=\sec^2 x-1\) to convert all but one tangent to secants. Then substitute \(u=\sec x,du=\sec x\tan x\,dx\).
Example Find \(\int\tan^3\theta\sec^3\theta\,d\theta\).
If secant’s power is even, use the identity \(\sec^2 x=\tan^2 x+1\) to convert all but two secants to tangents. Then substitute \(u=\tan x,du=\sec^2 x\,dx\).
Example Find \(\int\sec^4 x\tan^5 x\,dx\).
Other examples, such as \(\int \sec^3 x\,dx\), require other techniques, such as integration by parts.

Find \(\int\sin^4 x\cos^3 x\,dx\).
Find \(\int\sin^5 \theta\cos^2 \theta\,d\theta\).
Find \(\int\sin^2 x\,dx\).
Find \(\int\cos^4 y\,dy\).
Find \(\int\tan^2 t\sec^4 t\,dt\).
Find \(\int\tan z\sec^5 z\,dz\).
Find \(\int\tan^3 x\,dx\). (Hint: rewrite as \(\int\tan^2 x\tan x\,dx\) and use a trig identity on \(\tan^2 x\). Also, recall that \(\int\tan x\,dx=\ln|\sec x|+C\).)
Find \(\displaystyle\int\frac{\sin^3 r}{\cos^7 r}\,dr\). (Hint: rewrite with tangent and secant.)
Show that \(\int_0^{\pi}\sqrt{2-2\cos y}\,dy=4\). (Hint: replace \(x\) with \(y/2\) in \(\sin^2(x)=\frac{1}{2}-\frac{1}{2}\cos(2x)\).)