Mathematics

# Evaluate the given integral.$\displaystyle\int { \cfrac { { e }^{ \tan ^{ -1 }{ x } } }{ 1+{ x }^{ 2 } } } dx$

##### SOLUTION
Let $t={\tan}^{-1}{x}\Rightarrow\,dt=\dfrac{1}{1+{x}^{2}}dx$

$I=\displaystyle\int{\dfrac{{e}^{{\tan}^{-1}{x}}}{1+{x}^{2}}dx}$

$=\displaystyle\int{{e}^{t}dt}$

$={e}^{t}+c$     .........where $c$ is the constant of integration

$={e}^{{\tan}^{-1}{x}}+c$

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

#### Realted Questions

Q1 Subjective Medium
Evaluate
$\int \sqrt{1+t^{3}} dt$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
The minimum value of the function f(x) = $\int^x_0 \frac{d \theta}{cos \theta} + \int^{\pi/2}_x \frac{d \theta}{sin \theta}$ where $x \in [0, \frac{\pi}{2}],$ is
• A. $ln(2\sqrt{2} + 2)$
• B. $ln(\sqrt{3} + 2)$
• C. $ln(\sqrt{2} + 3)$
• D. $2ln(\sqrt{2} + 1)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium

Evaluate the given integral
$\displaystyle \int_{-1}^{1}xe^{x}dx=$
• A. $e$
• B. $\dfrac{1}{e}$
• C. ${e}^{2}$
• D. $\dfrac{2}{e}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\int {\frac{{dx}}{{\left( {4 + 3{x^2}} \right)\sqrt {3 - 4{x^2}} }} = }$
• A. $\frac{1}{5}{\tan ^{ - 1}}\frac{{2x}}{{\sqrt {3 - 4{x^2}} }} + c$
• B. $\frac{1}{{10}}{\tan ^{ - 1}}\frac{{5x}}{{2\sqrt {3 - 4{x^2}} }} + c$
• C. $\frac{1}{5}{\tan ^{ - 1}}\frac{{5x}}{{2\sqrt {3 - 4{x^2}} }} + c$
• D. $\frac{1}{{10}}{\tan ^{ - 1}}\frac{{5x}}{{\sqrt {3 - 4{x^2}} }} + c$

Integrate the following functions w.r.t $X :\dfrac{1}{\sqrt x+ \sqrt{x^3}}$