Mathematics

# Prove that $\displaystyle \int {\dfrac{{{{\sin }^{ - 1}}x}}{{{{\left( {1 - {x^2}} \right)}^{\dfrac{3}{2}}}}}} dx$

##### SOLUTION
$I=\int \dfrac{\sin^{-1}x}{(1-x^{2})^{\frac{3}{2}}}dx$

Let $\sin^{-1}x=u\Rightarrow \dfrac{1}{\sqrt{1-x^{2}}}dx=du$

So : $\displaystyle\int \dfrac{\sin^{-1}x}{(1-x^{2})(1-x^{2})^{\dfrac{1}{2}}}dx=\int \frac{u}{(1-\sin^{2}u)}du$

$= \displaystyle\int \dfrac{u}{\cos^{2}u}dx=\int u \,cosec^{2}u\, du$

$\Rightarrow u\displaystyle\int cosec^{2}u\,du-\int \left(\int cosec^{2}u\,du\right)du$

$=-u \cot u +\int \cot u \,du$

$=-u\cot u+\int \cot u\, du$

$=-u \cot u+In \sin u+a$

$=-u \cot u+In \sin u+a$

$I=-\sin^{-1}x\cot \sin^{-1}u+ ln x+a$

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Subjective Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Matrix Medium
Match the correct pair.
 $\int \dfrac {dx}{x^{2} + a^{2}}$ $\log [x - \sqrt {x^{2} - a^{2}}]$ $\int \dfrac {dx}{\sqrt {a^{2} - x^{2}}}$ $\dfrac {1}{2} x\sqrt {a^{2} - x^{2}} + \dfrac {1}{2}a^{2}\sin^{-1} \dfrac {x}{a}$ $\int \sqrt {a^{2} - x^{2}} dx$ $\dfrac {1}{a} \tan^{-1} \left (\dfrac {x}{a}\right )$ $\int \dfrac {dx}{\sqrt {x^{2} - a^{2}}}$ $a.\tan^{-1}x$ $\int \sqrt {a^{2} + x^{2}}dx$ $\sin^{-1}\left (\dfrac {x}{a}\right )$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\displaystyle\int_{\frac{1}{9}}^{\frac{1}{4}} {\dfrac{{dx}}{{(1 - x)\sqrt x }}} = \,\log \dfrac{3}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove that $\displaystyle\int^{3a/4}_{a/4}\dfrac{\sqrt{x}}{(\sqrt{a-x}+\sqrt{x})}dx=\dfrac{a}{4}$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle I=\int \frac{dx}{x^{3}\sqrt{x^{2}-1}}$, then I equals
• A. $\displaystyle \frac{1}{2}\left ( \frac{\sqrt{x^{2}-1}}{x^{3}}+\tan^{-1}\sqrt{x^{2}-1} \right )+C$
• B. $\displaystyle \frac{1}{2}\left ( \frac{\sqrt{x^{2}-1}}{x^{2}}+x\tan^{-1}\sqrt{x^{2}-1} \right )+C$
• C. $\displaystyle \frac{1}{2}\left ( \frac{\sqrt{x^{2}-1}}{x}+\tan^{-1}\sqrt{x^{2}-1} \right )+C$
• D. $\displaystyle \frac{1}{2}\left ( \frac{\sqrt{x^{2}-1}}{x^{2}}+\tan^{-1}\sqrt{x^{2}-1} \right )+C$

Solve $\int { { x }^{ 2 }. } \cos { \left( { x }^{ 3 } \right) } \sqrt { \sin ^{ 7 }{ \left( { x }^{ 3 } \right) } } dx$