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# Evaluate the following integral:$\int{\tan x{{\sec }^{2}}x\sqrt{1-{{\tan }^{2}}x}dx}$  Hint: To solve this question substitute value of $1-{{\tan }^{2}}x=t$
We have the given integral as $I=\int{\tan x{{\sec }^{2}}x\sqrt{1-{{\tan }^{2}}x}dx}..........\left( 1 \right)$

Here, we can use substitute method for finding/solving the given integral in a proper way:
Let $t=1-{{\tan }^{2}}x$
Differentiating both sides with respect to $x$
$t=1-{{\tan }^{2}}x$
$\dfrac{dt}{dx}=-2\tan x{{\sec }^{2}}x$ $\left( \dfrac{d}{dx}\left( \tan x \right)\ And -{{\sec }^{2}}x \right)$ chain rule is applied
$dt=-2\tan x{{\sec }^{2}}xdx.............\left( 2 \right)$
From the equation $\left( 1 \right)\And \left( 2 \right)$; we can replace $\tan x{{\sec }^{2}}xdx$ by
above equation $\left( 2 \right)$ as
$\tan x{{\sec }^{2}}xdx=\dfrac{-dt}{2}$
Hence, equation $\left( 1 \right)$ will become
$I=\int{\dfrac{-1}{2}\sqrt{t}dt}$ as $\left( 1-{{\tan }^{2}}x=t \right)$
\begin{align} & I=\dfrac{-1}{2}\int{{{t}^{\dfrac{1}{2}}}}dt \\ & I=\dfrac{-1}{2}\dfrac{{{t}^{\dfrac{1}{2}+1}}}{\dfrac{1}{2}+1}+C\text{ } as \int{{{x}^{n}}dx=\dfrac{{{x}^{n+1}}}{n+1}} \\ \end{align}
$I=\dfrac{-1}{2}\dfrac{{{t}^{\dfrac{3}{2}}}}{\dfrac{3}{2}}+C=\dfrac{-1}{2}\times \dfrac{2}{3}{{t}^{\dfrac{3}{2}}}+C$
\begin{align} & I=\dfrac{-1}{3}{{t}^{\dfrac{3}{2}}}+C \\ & \text{As }t=1-{{\tan }^{2}}x \\ & I=\dfrac{-1}{3}{{\left( 1-{{\tan }^{2}}x \right)}^{\dfrac{3}{2}}}+C \\ \end{align}
Hence,
$\int{\tan x{{\sec }^{2}}x\sqrt{1-{{\tan }^{2}}x}}dx=\dfrac{-1}{3}{{\left( 1-{{\tan }^{2}}x \right)}^{\dfrac{3}{2}}}+C$

Note: One can substitute
$t=\tan x$
Hence $dt={{\sec }^{2}}xdx$ and then can put value in integral.
Therefore $I=\int{t\sqrt{1-{{t}^{2}}}dt}$
Now, he/she needs to put ${{t}^{2}}=y\And 1-{{t}^{2}}=y$ to solve the above integral.
Hence, it takes one more step than the solution provided but the answer will be the same.
One can convert $\tan x\And {{\sec }^{2}}x$ to cosine and sine forms which students do
generally will take more time as well.
View Notes
Integration by Substitution  Surface Integral  Line Integral  Double Integral  Riemann Integral  Integral Calculus  Definite Integral  Integral Test  Binomial Theorem for Positive Integral Indices  Integration by Parts Rule  