Integrating $\int \frac{x \cos x+1}{\sqrt{2x^3e^{\sin x}+x^2}}dx$

Question

I came across a question today..

Integrate $\int \dfrac{x \cos x+1}{\sqrt{2x^3e^{\sin x}+x^2}}dx$

How to do it? I tried to take $x^2 e^{\sin x}$ out of the roots. But it didn't work out. I also tried to used substitution method by with the whole denominator but no result.

Answer 1

Take $x e^{\sin x}$ =t/2 and you will get the answer. Multiply the numerator and the denominator by ${e^{\sin x}}$ and pull $x^2$ out of the root. $\int{dt\over t\sqrt{t+1}}$.Now substitute $t+1=y^{2}$.