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

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

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

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

$=\dfrac{u^5}5+C$

$=\boxed{\dfrac{(\cos\theta+\sin\theta)^5}5+C}$