Write a value of ^2x (5+ )^4 dx - Vidyadip

Question

Write a value of $\int\frac{\sec^2\text{x}}{(5+\tan\text{x})^4}\text{ dx}$

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Answer

Let $\text{I}=\int\frac{\sec^2\text{x}}{(5+\tan\text{x})^4}\text{ dx}$
Let $5+\tan\text{x}=\text{t}$
$\sec^2\text{x dx}=\text{dt}$
$\therefore\ \text{I}=\int\frac{\text{dt}}{\text{t}^4}$
$=-\frac{1}{3\text{t}^3}+\text{C}$
$\text{I}=-\frac{1}{3(5+\tan\text{x})^3}+\text{C}$

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