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.

  • $\begingroup$ 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. $\endgroup$ – DonAntonio Mar 1 '16 at 8:48
  • $\begingroup$ Is it a problem for students? What course? $\endgroup$ – Yuriy S Mar 1 '16 at 8:49
  • $\begingroup$ @YuriyS It's like high level high school problem.. :p $\endgroup$ – manshu Mar 1 '16 at 8:57
  • $\begingroup$ Depends on the country, I guess. We had nothing like this in high-school $\endgroup$ – Yuriy S Mar 1 '16 at 9:18

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

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

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