Question

Find the trigonometric integral. (Use C for the constant of integration.)

Answer 1

Suppose we let
`u=\sin\theta+\cos\theta`, so that
`\mathrm du=(\cos\theta-\sin\theta)\,\mathrm d\theta`.

Also, recall the double angle identity for cosine:

`\cos(2\theta)=\cos^2\theta-\sin^2\theta=(\cos\theta-\sin\theta)(\cos\theta+\sin\theta)`

So, we can rewrite and compute the integral using the substitution, as

`\displaystyle\int\cos(2\theta)(\sin\theta+\cos\theta)^3\,\mathrm d\theta`

`=\displaystyle\int u\cdot u^3\,\mathrm du`

`=\displaystyle\int u^4\,\mathrm du`

`=\frac{u^5}5+C`

`=\boxed{\frac{(\cos\theta+\sin\theta)^5}5+C}`