Write a value of ^3x dx - Vidyadip

Question

Write a value of $\int\tan\text{x}\sec^3\text{x dx}$

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Answer

Let $\text{I}=\int\tan\text{x}\sec^3\text{x dx}$
Let $\sec\text{x}=\text{t}$
$\sec\text{x}\tan\text{x dx}=\text{dt}$
$\text{dx}=\frac{\text{dt}}{\sec\text{x}\tan\text{x}}$
$\therefore\ \text{I}=\int\sec^2\text{x}\tan\text{x dx}$
$=\int\text{t}^2\text{ dt}$
$=\frac{\text{t}^3}{3}+\text{C}$
$=\frac{\sec^3\text{x}}{3}+\text{C}$
$\therefore\ \text{I}=\frac{\sec^3\text{x}}{3}+\text{C}$

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