Question

Integrate the function w..r. to x : $\sin\theta\cdotp\log (\cos\theta)$

Answer 1

This integration can be done by substitution then applying by parts

Let

$cos\:\theta = t\\\\ sin\:\theta\:d\theta= -dt$

After substituting ,we get

$-\int (log\: t) dt=log\: t\int 1 dt-\int (\frac{log \:t}{dt}\int 1 dt) dt\\\\= t \:log\: t -\int \frac{1}{t}.{t} dt\\\\= t \:log t-t +C\\\\\int log \:t dt = t - log \:t+C$

Now undo substitution

$\int sin\:\theta \:log(cos \:\theta) \:d\theta= cos\: \theta -log\:(cos \:\theta)+C$