Mathematics

# $\int {\frac{{{{\sin }^{ - 1}}x}}{{{{\left( {1 - {x^2}} \right)}^{\frac{3}{2}}}}}dx}$

$\frac{{x\left( {{{\sin }^{ - 1}}x} \right)}}{{\sqrt {1 - {x^2}} }} + \frac{1}{2}\log \left| {\left( {1 - {x^2}} \right)} \right| + C.$

##### SOLUTION
$\begin{array}{l} Put\, x=\sin t\, \, so,\, that\, dx={ { costdt } }\, and\, t=si{ n^{ -1 } }x \\ \therefore \int { \frac { { { { \sin }^{ -1 } }x } }{ { { { \left( { 1-{ x^{ 2 } } } \right) }^{ \frac { 3 }{ 2 } } } } } dx=\int { \frac { { t\cos t } }{ { { { \left( { 1-{ { \sin }^{ 2 } }t } \right) }^{ \frac { 3 }{ 2 } } } } } dt } =\int { \frac { { t{ { cost } } } }{ { { { \cos }^{ 3 } }t } } dt } } \\ =\int { t{ { \sec }^{ 2 } }tdt } \\ =t.\left( { \tan t } \right) -\int { 1.\tan tdt } \\ =t\left( { \tan t } \right) +\log \left| { \cos t } \right| +C \\ ={ \sin ^{ -1 } }x\frac { x }{ { \sqrt { 1-{ x^{ 2 } } } } } +\log \left| { \sqrt { 1-{ x^{ 2 } } } } \right| +C \\ =\frac { { x\left( { { { \sin }^{ -1 } }x } \right) } }{ { \sqrt { 1-{ x^{ 2 } } } } } +\frac { 1 }{ 2 } \log \left| { \left( { 1-{ x^{ 2 } } } \right) } \right| +C. \\ hence,\, the\, \, option\, \, A\, is\, correct\, \, answer. \end{array}$

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

#### Realted Questions

Q1 Subjective Hard
Evaluate : $\displaystyle\int \dfrac{\cos^{-1}x}{x^2} dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Evaluate: $\displaystyle\int { \dfrac { { x }^{ 2 }+1 }{ { x }^{ 4 }+1 } dx }$ equals
• A. $\dfrac{1}{\sqrt{2}}\tan^{-1}\left(\dfrac{1-x^{2}}{\sqrt{2}x}\right)+C$
• B. $\dfrac{1}{2}\tan^{-1}\left(\dfrac{x^{2}-1}{\sqrt{2}x}\right)+C$
• C. $\dfrac{1}{2}\tan^{-1}\left(\dfrac{1-x^{2}}{\sqrt{2}x}\right)+C$
• D. $\dfrac{1}{\sqrt{2}}\tan^{-1}\left(\dfrac{x^{2}-1}{\sqrt{2}x}\right)+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\displaystyle\int \sin^{\dfrac{5}{2}}x\cos^3xdx=?$
• A. $2\sin^{\dfrac{7}{2}}x\left[\dfrac{1}{7}-\dfrac{1}{11}\sin^2x\right]+c$
• B. $3\sin^{\dfrac{7}{2}}x\left[\dfrac{1}{5}-\dfrac{1}{9}\sin^2x\right]+c$
• C. $\sin^{\dfrac{7}{2}}x\left[\dfrac{1}{5}+\dfrac{1}{9}\sin^2x\right]+c$
• D. $2\sin^{\dfrac{7}{2}}x\left[\dfrac{1}{7}+\dfrac{1}{11}\sin^2x\right]+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Integrate the rational function: $\cfrac {1}{x^2-9}$

$\int(2x^2-3 \, sin x+5 \sqrt{x})dx$