Evaluate the following integrals: ^2x 1- ^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\tan\text{x}\sec^2\text{x}\sqrt{1-\tan^2\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int\tan\text{x}\sec^2\text{x}\sqrt{1-\tan^2\text{x}}\text{ dx}\ ....(1)$ Let $1-\tan^2\text{x}=\text{t}$ then, $\Rightarrow\text{d}\big(1-\tan^2\text{x}\big)\text{dt}$ $\Rightarrow-2\tan\text{x}\sec^2\text{x}\text{ dx}=\text{dt}$ $\Rightarrow\tan\text{x}\sec^2\text{x}\text{ dx}=\frac{-\text{dt}}{2}$Putting $1-\tan^2\text{x}=\text{t}$ and $\tan\text{x}\sec^2\text{x}\text{ dx}=-\frac{\text{dt}}{2}$ in equation (1),
We get
$\text{I}=\int\sqrt{\text{t}}\times\frac{-\text{dt}}{2}$
$=\frac{-1}{2}\int\text{t}^{\frac{1}{2}}\text{dt}$
$=-\frac{1}{2}\times\frac{\text{t}^{\frac{3}{2}}}{\frac{3}{2}}+\text{C}$
$=-\frac{1}{3}\text{t}^\frac{3}{2}+\text{C}$
$=-\frac{1}{3}\big[1-\tan^2\text{x}\big]^{\frac{3}{2}}+\text{C}$

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