Evaluate the following integrals: x x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\cos\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$

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Answer

$\int\frac{\cos\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$

Let $\sqrt{\text{x}}=\text{t}$

$\Rightarrow\frac{1}{2\sqrt{\text{x}}}=\frac{\text{dt}}{\text{dx}}$

$\Rightarrow\frac{\text{dx}}{\sqrt{\text{x}}}=2\text{dt}$

Now, $\int\frac{\cos\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$

$=2\int\cos\text{t dt}$

$=2\sin\text{t}+\text{C}$

$=2\sin\sqrt{\text{x}}+\text{C}$

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