Mathematics

# $\displaystyle \int \frac{x\: dx}{1+x^{4}}$ is equal to

$\displaystyle \frac{1}{2}\tan ^{-1}x^{2}+k$

##### SOLUTION
$\displaystyle \int \dfrac{x\: dx}{1+x^{4}}$

$\displaystyle =\int \dfrac { x\: dx }{ 1+{ (x }^{ 2 })^{ 2 } }$

Put ${ x }^{ 2 }=t$
$\Rightarrow 2xdx=dt$

$\displaystyle =\dfrac { 1 }{ 2 } \int \dfrac { dt }{ 1+t^{ 2 } }$

$\displaystyle =\dfrac { 1 }{ 2 } \tan ^{ -1 }{ t } +k$

$I=\displaystyle \dfrac{1}{2}\tan ^{-1}x^{2}+k$

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

#### Realted Questions

Q1 Subjective Hard
Evaluate $\displaystyle\int _{ 0 }^{ \pi }{ \dfrac { x\sin ^{ 3 }{ x } }{ 1+\cos ^{ 2 }{ x } } dx }$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Assertion & Reason Hard
##### ASSERTION

Assume: $\displaystyle I=\int \frac{\sqrt{\cos 2x}}{\sin x}dx$

$\displaystyle I=\ln \left [ \left ( \frac{1-\sqrt{1-\tan ^{2}x}}{\tan x} \right )\left ( \frac{\sqrt{2}+\sqrt{1-\tan ^{2}x}}{\sqrt{2}-\sqrt{1-\tan ^{2}x}} \right )^{\frac{1}{\sqrt{2}}} \right ]+C$

##### REASON

$\displaystyle \tan x=\sin \theta \rightarrow I=\int \frac{\cos ^{2}\theta d\theta }{\sin \theta \left ( 1+\sin ^{2}\theta \right )}$

• A. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Both Assertion and Reason are incorrect
• D. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Solve $\displaystyle\int \sqrt{2+\sin 3x}\cos 3xdx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Value of $I= \displaystyle \int_{0}^{a}\displaystyle \frac{dx}{x+\sqrt{a^{2}-x^{2}}}$ is
• A. $\pi /2$
• B. $\pi$
• C. $2\pi$
• D. $\pi /4$

Let g(x) =$\displaystyle \int_{0}^{x}f\left ( t \right )dt,$ where f is a function