Mathematics

# $\displaystyle \int \dfrac x{2+x^2} dx$

##### SOLUTION
$\displaystyle \int \dfrac x{2+x^2} dx\\2+x^2=t\implies 2xdx=dt\\\displaystyle \int \dfrac 1{2t} dt\\\dfrac 12\log t\\\dfrac 12{\log(2+x^2)}+c$

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

#### Realted Questions

Q1 Single Correct Medium
If $\int { \cfrac { 1-{ \left( \cot { x } \right) }^{ 2010 } }{ \tan { x } +{ \left( \cot { x } \right) }^{ 2011 } } dx } =\cfrac { 1 }{ k } \log _{ e }{ \left| { \left( \sin { x } \right) }^{ k }+{ \left( \cos { x } \right) }^{ k } \right| } +C$, then $k$ is equal to
• A. $2010$
• B. $2011$
• C. $2013$
• D. $2012$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int x\{f(x^{2}) g' (x^{2})+f '(x^{2})g(x^{2})\}dx$ is equal to
• A. $f(x^{2}) g$' $(x^{2})-g(x^{2}) f$'$(x^{2})+c$
• B. ${\dfrac{1}{2}}\{f(x^{2})g(x^{2}) f$' $(x^{2})\}+c$
• C. ${\dfrac{1}{2}}\{f(x^{2}) g$' $(x^{2})-g(x^{2}) f$' $(x^{2})\}+c$
• D. ${\dfrac{1}{2}} [f(x^{2})g(x^{2})]+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Matrix Hard
The value of $\displaystyle I = \int_{0}^{a}f\left ( x \right )dx$
 $a= \pi ,f\left ( x \right )= x\sin ^{4}x$ $\pi \left ( \pi -2 \right )/2$ $a= \pi ,f\left ( x \right )= x\sin ^{4}x\cos ^{6} x$ $\displaystyle \frac{\pi }{3\sqrt{3}}$ $a= \pi/2 ,f\left ( x \right )= \displaystyle \frac{\cos ^{2} x}{1+\cos x\sin x}$ $\displaystyle \frac{3\pi^{2} }{512}$ $a= \pi ,f\left ( x \right )= \displaystyle \frac{x\tan x}{\sec x+\tan x}$ $\displaystyle \frac{3}{16}\pi^{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Solve: $\displaystyle\int \dfrac {dx}{\sqrt {21-4x-x^2}}$
• A. $\arcsin \dfrac{(x+4)}{5} +C$
• B. $\arcsin \dfrac{(x-4)}{5} +C$
• C. None of these
• D. $\arcsin \dfrac{(x+2)}{5} +C$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
The value of $\int _{ { \pi }/{ 4 } }^{ { \pi }/{ 2 } }{ { e }^{ x }\left( \log { \sin { x } } +\cot { x } \right) dx }$ is
• A. ${ e }^{ { \pi }/{ 4 } }\log { 2 }$
• B. $-{ e }^{ { \pi }/{ 4 } }\log { 2 }$
• C. $-\dfrac { 1 }{ 2 } { e }^{ { \pi }/{ 4 } }\log { 2 }$
• D. $\dfrac { 1 }{ 2 } { e }^{ { \pi }/{ 4 } }\log { 2 }$