Evaluate the following integrals: ^2 x x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\sec^2\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$

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Answer

$\int\frac{\sec^2\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$

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, $\int\frac{\sec^2\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$

$=2\int\sec^2\text{t dt}$

$=2\tan(\text{t})+\text{C}$

$=2\tan\big(\sqrt{\text{x}}\big)+\text{C}$

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