[Math] Solve $\int e^{\cos x}\frac{x \sin^3x+\cos x}{\sin^2x}dx$

Question

$$\int e^{\cos x} \left(\frac{x \sin^3x+\cos x}{\sin^2x}\right)dx$$

At first look, i thought $\int e^x(f(x)+f'(x))dx$ will be applied but not applicable in this case.
Then i used integration by parts but that is not working either.How should i solve it,please help…

Answer 1

Substitute $t=\cos(x)$ to get

$$\int e^t \left(\arccos(t)-\frac t{(1-t^2)^{3/2}}\right)dt$$

and apply your technique with a plus/minus trick:

$$\int e^t \left(\arccos(t)-\frac1{\sqrt{1-t^2}}+\frac1{\sqrt{1-t^2}}-\frac t{(1-t^2)^{3/2}}\right)dt=e^t\arccos(t)+\frac{e^t}{\sqrt{1-t^2}}.$$