Question

Evaluate:
$\int \left(\frac{x\cos x+\sin x}{x\sin x}\right)dx$

Answer 1

Consider the given integral.

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

Let

$t=x\sin x$

$\frac{dt}{dx}=x\cos x+\sin x\times 1$

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

Therefore,

$I=\int \frac{1}{t}dt$

$I=\ln t+C$

On putting the value of the $t$, we get

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

Hence, this is the answer.