Mathematics

# $\displaystyle\int _{ 0 }^{ \infty }{ \dfrac { dx }{ \left( x+\sqrt { { x }^{ 2 }+1 } \right) ^{ 3 } } } =$

$\dfrac{3}{8}$

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

#### Realted Questions

Q1 Single Correct Medium

$\displaystyle \int_{0}^{1}\frac{x}{1+\sqrt{x}}dx_{=}$
• A. $\frac{5}{3}+\log 4$
• B. $\frac{5}{3}\log 4$
• C. $\displaystyle \frac{3}{5}-l\mathrm{o}\mathrm{g}4\frac{3}{5}-l\mathrm{o}\mathrm{g}4$
• D. $\frac{5}{3}-\log 4$

1 Verified Answer | Published on 17th 09, 2020

Q2 One Word Medium
If $\displaystyle \int \left[\left(\dfrac{x}{e}\right)^x + \left(\dfrac{e}{x} \right)^x \right] \ln x \, dx = A \left(\dfrac{x}{e}\right)^x + B \left(\dfrac{e}{x} \right)^x + C$, then value of $A + B =$?

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Hard
Suppose $I_1=\displaystyle \int_{0}^{\pi/2} \cos(\pi \sin^2 x)dx;I_2=\displaystyle \int_{0}^{\pi/2} \cos(2\pi \sin^2x)dx$ and $I_3=\displaystyle \int_{0}^{\pi/2} \cos(\pi \sin x)dx$ then
• A. $I_2=I_3$
• B. $I_1=0$
• C. $I_2+I_3=0$
• D. $I_1+I_2+I_3=0$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int\frac{x\sin^{-1}x}{\sqrt{1-x^{2}}}dx$ is equal to
• A. $x+\sqrt{1-x^{2}}sin^{-1}x+c$
• B. $x+sin^{-1}x+c$
• C. $x-sin^{-1}x+c$
• D. $x-\sqrt{1-x^{2}}sin^{-1}x+c$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Hard
If $\displaystyle I=\int _{ 0 }^{ { \pi }/{ 2 } }{ \ell n { \left( \sin { x } \right) } }\ dx$ then $\displaystyle \int _{ { -\pi }/{ 4 } }^{ { \pi }/{ 4 } }{ \ell n { \left( \sin { x } +\cos { x } \right) dx= } }$
• A. $\dfrac { l }{ 4 }$
• B. $\dfrac { l }{ \sqrt { 2 } }$
• C. $I$
• D. $\dfrac { l }{ 2 }$