Indefinite integral of $\cos^{3}(x) \cdot \ln(\sin(x))$

Question

I need help. I have to integrate $\cos^{3} \cdot \ln(\sin(x))$ and I don´t know how to solve it. In our book it is that we have to solve using the substitution method. If somebody knows it, you will help me..please

Answer 1

Substitute :

$\sin x =t \Rightarrow \cos x dx =dt$ , hence :

$I=\int (1-t^2)\cdot \ln (t) \,dt$

This integral you can solve using integration by parts method .

Answer 2

If you mean $\cos^3 x\ln(\sin x)$, let $u=\sin x$. Then $du=\cos x dx$, and $$\begin{align*} \cos^3 x\ln(\sin x)dx&=\cos^2 x\ln(\sin x)\Big(\cos x dx\Big)\\ &=\cos^2 x\ln u \,du\\ &=(1-\sin^2 x)\ln u\,du\\ &=(1-u^2)\ln u\,du\;, \end{align*}$$

which can be integrated by parts.