Mathematics

# Evaluate the given integral.$\displaystyle\int { \cfrac { x }{ 4+{ x }^{ 4 } } } dx$

$\cfrac { 1 }{ 4 } \tan ^{ -1 }{ \cfrac { { x }^{ 2 } }{ 2 } }$

##### SOLUTION
Let $u={x}^{2}\Rightarrow\,du=2x\,dx$

$I=\displaystyle\int{\dfrac{x\,dx}{4+{x}^{4}}}$

$=\dfrac{1}{2}\displaystyle\int{\dfrac{du}{4+{u}^{2}}}$

$=\dfrac{1}{2}\times\dfrac{1}{2}{\tan}^{-1}{\left(\dfrac{u}{2}\right)}+c$    .......where $c$ is the constant of integration.

$=\dfrac{1}{4}{\tan}^{-1}{\left(\dfrac{{x}^{2}}{2}\right)}+c$    ......where $u={x}^{2}$

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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 Hard
If $0< \alpha < \pi /2$ then the value of $\displaystyle \int_{0}^{\alpha }\displaystyle \frac{dx}{1-\cos x\cos \alpha }$ is
• A. $\pi /\alpha$
• B. $\pi /2\cos \alpha$
• C. $\pi /2\alpha$
• D. $\pi /2\sin \alpha$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\int^{4}_{1}\left(x^{2}-x\right)dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The value of the integral $\int _{ 1 }^{ { 3 }^{ 1/n } }{ \cfrac { dx }{ x({ x }^{ n }+1) } }$ is
• A. $\cfrac { 1 }{ n } \log { \left( \cfrac { 2 }{ 3 } \right) }$
• B. $n\log { \left( \cfrac { 2 }{ 3 } \right) }$
• C. $n\log { \left( \cfrac { 3 }{ 2 } \right) }$
• D. $\cfrac { 1 }{ n } \log { \left( \cfrac { 3 }{ 2 } \right) }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Evaluate the integral
$\displaystyle \int_{-\pi}^{\pi}\left ( x^{3}+x\cos x+tan^{5}x+2 \right )dx$
• A. $0$
• B. $\pi$
• C. $2\pi$
• D. $4\pi$

$\displaystyle\int x^n\log_ex\ dx$