Question:
Evaluate the following integrals:
$\int \frac{e^{\sqrt{x}} \cos \left(e^{\sqrt{x}}\right)}{\sqrt{x}} d x$
Solution:
Assume $e^{\sqrt{x}}=t$
$d\left(e^{\sqrt{x}}\right)=d t$
$\Rightarrow \frac{e^{\sqrt{x}}}{2 \sqrt{x}} d x=d t$
$\Rightarrow \frac{e^{\sqrt{x}}}{\sqrt{x}} d x=2 d t$
Substituting $\mathrm{t}$ and $\mathrm{dt}$
$\Rightarrow 2 \int \cos t d t$
$=2 \sin t+c$
But $t=e^{\sqrt{x}}$
$\Rightarrow 2 \sin \left(e^{\sqrt{x}}\right)+c$
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