Evaluate ^2 x x dx - Vidyadip

Question

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

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Answer

Let $\text{I}=\int\frac{\sec^2\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$
Let $\sqrt{\text{x}}=\text{t}$
$\frac{\text{dx}}{2\sqrt{\text{x}}}=\text{dt}$
$\frac{\text{dx}}{\sqrt{\text{x}}}=2\text{dt}$
Putting $\sqrt{\text{x}}=\text{t}$ and $\frac{\text{dx}}{\sqrt{\text{x}}}=2\text{dt}$
$\therefore\ \text{I}=2\int\sec^2+\text{dt}$
$=2\tan\text{t}+\text{C}$
$=2\tan(\sqrt{\text{x}})+\text{C}$ $(\because\text{t}=\sqrt{\text{x}})$

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