Evaluate the following integrals: e^ x (e^ x ) x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{e}^{\sqrt{\text{x}}}\cos\big(\text{e}^{\sqrt{\text{x}}}\big)}{\sqrt{\text{x}}}\text{ dx}$

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Answer

$\int\frac{\text{e}^{\sqrt{\text{x}}}\cos\big(\text{e}^{\sqrt{\text{x}}}\big)}{\sqrt{\text{x}}}\text{ dx}$
Let $\text{e}^{\sqrt{\text{x}}}=\text{t}$
$\Rightarrow\text{e}^{\sqrt{\text{x}}}\times\frac{1}{2\sqrt{\text{x}}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\frac{\text{e}^{\sqrt{\text{x}}}}{\sqrt{\text{x}}}\text{ dx}=2\text{dt}$
Now, $\int\frac{\text{e}^{\sqrt{\text{x}}}\cos\big(\text{e}^{\sqrt{\text{x}}}\big)}{\sqrt{\text{x}}}\text{ dx}$
$=2\int\cos\text{t}\text{ dt}$
$=2\sin\text{t}+\text{C}$
$=2\sin\Big(\text{e}^\sqrt{\text{x}}\Big)+\text{C}$

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