# Section 2.2 Calculus 2

Integration of Trigonometric Products

## 2.2 Integration of Trigonometric Products

### 2.2.1 Integrating Products of Sine and Cosine

• 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 $$\int\sin^2 z\cos^2 z\,dz$$.

### 2.2.2 Integrating Products of Secant and Tangent

• 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.

### Review Exercises

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

Solutions 1-5

Solutions 6-9

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 8.2