Question

How to solve this problem using integration?

Answer 1

$\displaystyle\int\cos^56x\sin6x\,\mathrm dx$

Let $y=\cos6x$, so that $\mathrm dy=-6\sin6x\,\mathrm dx$. Then

$\displaystyle\int\cos^56x\sin6x\,\mathrm dx=-\frac16\int y^5\,\mathrm dy=-\frac1{36}y^6+C=-\frac1{36}\cos^66x+C$