Mathematics

# $\displaystyle \int { \cfrac { x\tan ^{ -1 }{ x } }{ { \left( 1+x^{ 2 } \right) }^{ 3/2 } } } dx=$

$\cfrac { x-\tan ^{ -1 }{ x } }{ { \sqrt { \left( 1+x^{ 2 } \right) } }^{ } } +c$

##### SOLUTION
Let $I = \displaystyle \int \dfrac{x \tan^{-1} x}{(1 + x^2)^{3/2}} dx$         let $x = \tan \theta$
$dx = \sec^2 \theta d \theta$

$I = \displaystyle \int \dfrac{\tan \theta . \theta . \sec^2 \theta}{\sec^3 \theta} d \theta = \int \dfrac{\theta \tan \theta }{\sec \theta } d \theta$

$I = \displaystyle \int \theta. \sin \theta \, d \theta = - \theta \cos \theta + \int \cos \theta$

$= - \theta \cos \theta + \sin \theta$

$= \dfrac{-\tan^{-1} x}{\sqrt{1 + x^2}} + \dfrac{x}{\sqrt{1 + x^2}} = \dfrac{x - \tan^{-1} x}{\sqrt{1 + x^2}} + C$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Hard
If $\displaystyle\int { { x }^{ 13/2 }.{ \left( 1+{ x }^{ 5/2 } \right) }^{ 1/2 }dx } =P{ \left( 1+{ x }^{ 5/2 } \right) }^{ 7/2 }+Q{ \left( 1+{ x }^{ 5/2 } \right) }^{ 5/2 }+R{ \left( 1+{ x }^{ 5/2 } \right) }^{ 3/2 }+C$, then $P,\ Q$ and $R$ are
• A. $P=\frac { 4 }{ 35 } ,\ Q=\frac { 8 }{ 25 } ,\ R=\frac { 4 }{ 15 }$
• B. $P=-\frac { 4 }{ 35 } ,\ Q=-\frac { 8 }{ 25 } ,\ R=\frac { 4 }{ 15 }$
• C. $P=\frac { 4 }{ 35 } ,\ Q=-\frac { 8 }{ 25 } ,\ R=-\frac { 4 }{ 15 }$
• D. $P=\frac { 4 }{ 35 } ,\ Q=-\frac { 8 }{ 25 } ,\ R=\frac { 4 }{ 15 }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the following definite integral.

$\displaystyle \int _{2}^3 \dfrac {x}{x^2+1}dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate $\displaystyle \int_{0}^{\pi/2} cos \,x \,e^{sin \,x} \,dx$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the integral $\displaystyle\int_{-4}^{4}|x+2|\ dx$.

Let $\displaystyle I_{1}=\int_{0}^{1}(1-x^{2})^{1/3} dx$  &  $\displaystyle I_{2}=\int_{0}^{1}(1-x^{3})^{1/2} dx$