Mathematics

# Solve $\displaystyle\int \dfrac{\sqrt{1+x^2}}{x^2}dx$

##### SOLUTION

Given: $\displaystyle\int \dfrac{\sqrt{1+x^2}}{x^2}dx$

Apply the integration by parts,

$u = \sqrt{1 + x^{2}}, v = \dfrac{1}{x^{2}}$

$\displaystyle\int \dfrac{\sqrt{1+x^2}}{x^2}dx = -\dfrac{\sqrt{1+x^2}}{x}-\int \:-\dfrac{1}{\sqrt{1+x^2}}dx$

$= -\dfrac{\sqrt{1+x^2}}{x}-\left(-\ln \left|\sqrt{x^2+1}+x\right|\right)$

$= -\dfrac{\sqrt{1+x^2}}{x}+\ln \left|\sqrt{x^2+1}+x\right|+C$

Hence, the required result is found.

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

#### Realted Questions

Q1 Single Correct Medium
The value of integral ${int}_{\pi/4}^{3\pi/4}\dfrac{x}{1+\sin x}dx$ is :
• A. $\pi\left(\sqrt{2}-1\right)$
• B. $\dfrac{\pi}{2}\left(\sqrt{2}+1\right)$
• C. $2\pi\left(\sqrt{2}-1\right)$
• D. $\pi\sqrt{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Evaluate $\int (\log x)^{2}dx$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\int_0^\pi {\frac{{dx}}{{1 + {2^{\tan x}}}}}$
• A. 2
• B. 3.62
• C. 4
• D. 1.57

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Q4 Subjective Medium
Evaluate: $\displaystyle\int \dfrac{3x^2}{x^6+1}$

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