Mathematics

# If $\int \frac { x ^ { 2 } \tan ^ { - 1 } x } { 1 + x ^ { 2 } } d x = \tan ^ { - 1 } x - \frac { 1 } { 2 } \log \left( 1 + x ^ { 2 } \right) + f ( x ) + c$ then $f ( x ) =$

$- \frac { 1 } { 2 } \left( \tan ^ { - 1 } x \right) ^ { 2 }$

##### SOLUTION

$\\\int\>(\frac{(1+x^2-1)tan^{-1}x}{1+x^2})dx\\=\int\>\left(1-(\frac{1}{1+x^2})\right)tan^{-1}x\>dx\\=\int\>1.tan^{-1}dx-\int\>(\frac{tan^{-1}x}{1+x^2})dx\\Use\>ILATE\>in\>first\>part\>and\>assume\\tan^{-1}x=t\>for\>second\>integration\\=xtan^{-1}x-\int\>(\frac{1}{1+x^2})\times\>xdx-\int\>t\>dt\\=xtan^{-1}x-(\frac{1}{2})log|1+x^2|-(\frac{t^2}{2})+C\\=xtan^{-1}x-(\frac{1}{2})log|1+x^2|-(\frac{(tan^{-1}x)^2}{2})+C\\\therefore\>C=-(\frac{1}{2})(tan^{-1}x)^2$

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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
Evaluate $\displaystyle\int_{\displaystyle\sqrt{\frac{(3a^2+b^2)}{4}}}^{\displaystyle\sqrt{\frac{(a^2+b^2)}{2}}}{\frac{x}{\sqrt{(x^2-a^2)(b^2-x^2)}}dx}$.
• A. $\displaystyle I=\frac{\pi}{6}$
• B. $\displaystyle I=\frac{\pi}{4}$
• C. $\displaystyle I=\frac{\pi}{2}$
• D. $\displaystyle I=\frac{\pi}{12}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int_0^a \dfrac{x^4}{a^2+x^2)^4}dx$
• A. $\dfrac{1}{4a^3}\left(\dfrac{\pi}{4}+\dfrac{1}{3}\right)$
• B. $\dfrac{1}{16a^3}\left(\dfrac{\pi}{4}-\dfrac{1}{3}\right)$
• C. $\dfrac{1}{4a^3}\left(\dfrac{\pi}{4}-\dfrac{1}{3}\right)$
• D. $\dfrac{1}{16a^3}\left(\dfrac{\pi}{4}+\dfrac{1}{3}\right)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
Solve : $\displaystyle \int \frac{(x+1)(x+ log x)^2}{x} dx$
• A. (b)$\dfrac{1}{3} (x + log x)^2 +c$
• B. (c)$\dfrac{1}{3} (x + log x)^3 +c$
• C. (d)None of these
• D. (a)$\dfrac{1}{3} (x + log x)^3 +c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\int \frac{1}{\sqrt{\left ( 1-4x-x^{2} \right )}}dx$
• A. $\displaystyle \sin ^{-1}\frac{x+2}{{\left ( 5 \right )}}$
• B. $\displaystyle \sin ^{-1}\frac{x-2}{\sqrt{\left ( 5 \right )}}$
• C. $\displaystyle \cos ^{-1}\frac{x+2}{\sqrt{\left ( 5 \right )}}$
• D. $\displaystyle \sin ^{-1}\frac{x+2}{\sqrt{\left ( 5 \right )}}$

$\int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx}$