Mathematics

# Integrate:$\displaystyle \dfrac {x\cdot \sec^{2}(x^{2})}{\sqrt {\tan^{3}(x^{2})}}$

##### SOLUTION

Consider the given integral.

$I=\displaystyle \int{\dfrac{x{{\sec }^{2}}\left( {{x}^{2}} \right)}{{{\tan }^{3}}\left( {{x}^{2}} \right)}dx}$

Let $t=\tan \left( {{x}^{2}} \right)$

$\dfrac{dt}{dx}={{\sec }^{2}}\left( {{x}^{2}} \right)\times 2x$

$\dfrac{dt}{2}=x{{\sec }^{2}}\left( {{x}^{2}} \right)dx$

Therefore,

$I=\dfrac{1}{2}\int{\dfrac{1}{{{t}^{3}}}dt}$

$I=\dfrac{1}{2}\left( -\dfrac{1}{2{{t}^{2}}} \right)+C$

$I=-\dfrac{1}{4{{t}^{2}}}+C$

On putting the value of $t$, we get

$I=-\dfrac{1}{4{{\left( \tan \left( {{x}^{2}} \right) \right)}^{2}}}+C$

$I=-\dfrac{1}{4{{\tan }^{2}}\left( {{x}^{2}} \right)}+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

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$\displaystyle \int\frac{\cos x+3\sin x+7}{\cos x+\sin x+1}dx$ is equal to
• A. $\displaystyle \log|\cos x+\sin x+1|+2x+5\log|1+\tan\frac{x}{2}|+c$
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• C. $-\log|\cos x +\sin_{X} +1|+2x -5\displaystyle \log|1+\tan\frac{x}{2}|+c$
• D. $-\log|\cos x+\sin x +1|+2x +5\displaystyle \log|1+\tan\frac{x}{2}|+c$

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Q2 Single Correct Hard
$\displaystyle\int\limits_{2 - \ell n3}^{3 + \ell n3} {\dfrac{{\ell n(4 + x)}}{{\ell n(4 + x) + \ell n(9 - x)}}dx}$ is  equal to:
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• C. is equal to $1 + 2\ell n3$
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1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
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