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

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.

• I'd love to see how to solve such an antiderivative. WA has no idea, and neither do I, what to do. It looks evil. – DonAntonio Mar 1 '16 at 8:48
• Is it a problem for students? What course? – Yuriy S Mar 1 '16 at 8:49
• @YuriyS It's like high level high school problem.. :p – manshu Mar 1 '16 at 8:57
• Depends on the country, I guess. We had nothing like this in high-school – Yuriy S Mar 1 '16 at 9:18

## 1 Answer

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 $t+1=y^{2}$.

• Excellent! It really works. +1 – DonAntonio Mar 1 '16 at 8:56
• Yes, I had my doubt's when reading this but now I've tried it, I'd say this was amazing. (+1) – Nikunj Mar 1 '16 at 9:12