Question

Find the indefinite integral. (use c for the constant of integration.) sin3 5θ cos 5θ dθ

Answer 1

I'm guessing you mean

$\displaystyle\int\sin^35\theta\cos5\theta\,\mathrm d\theta$

Let $t=5\sin\theta$. Then $\mathrm dt=5\cos5\theta\,\mathrm d\theta$. So we have

$\displaystyle\int\sin^35\theta\cos5\theta\,\mathrm d\theta=\frac15\int\sin^35\theta(5\cos5\theta)\,\mathrm d\theta$

$=\displaystyle\frac15\int t^3\,\mathrm dt$

$=\dfrac{t^4}{20}+C$

$=\dfrac{\sin^4\5\theta}{20}+C$