Question

Evaluate:

$\int \left(\frac{x\sin x-\cos x}{x\cos x}\right)dx$

Answer 1

We have,

$I=\int \left(\frac{x\sin x-\cos x}{x\cos x}\right)dx$

Let

$t=x\cos x$

$\frac{dt}{dx}=\cos x-x\sin x$

$dt=-(x\sin x-\cos x)dx$

Therefore,

$I=-\int \left(\frac{1}{t}\right)dt$

$I=-\ln t+C$

Put the value of $t$, we get

$I=-\ln \left(x\cos x\right)+C$

Hence, this is the answer.