Question

Solve $\int \frac{e^{\sqrt{x}}cos\left(e^{\sqrt{x}}\right)}{\sqrt{x}}dx$

Answer 1

$\int \frac{e^{\sqrt{x}}\cos \left(e^{\sqrt{x}}\right)}{\sqrt{x}}dt$

Let $e^{\sqrt{x}}=t{e}^{\sqrt{x}}.\frac{1}{2\sqrt{x}}dx=dt$

$=\int \frac{2e^{\sqrt{x}}\cos \left(e^{\sqrt{x}}\right)}{2\sqrt{x}}dx=2\int \cos tdt=2\sin t+C=2\sin \left(e^{\sqrt{x}}\right)+C$