Mathematics

# Evaluate the following integral:$\displaystyle \int { \cfrac { 1 }{ { e }^{ x }+{ e }^{ -x } } } dx$

##### SOLUTION
Consider $I=\displaystyle\int{\dfrac{dx}{{e}^{x}+{e}^{-x}}}$

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

Let $\tan{t}={e}^{x}\Rightarrow\,{\sec}^{2}{t}dt={e}^{x}dx$

$\Rightarrow\,\dfrac{{\sec}^{2}{t}}{2\tan{t}}dt=dx$

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

$I=\displaystyle\int{\dfrac{{\sec}^{2}{t}dt}{{\sec}^{2}{t}}}$

$I=\displaystyle\int{dt}$

$I=t+c$

$\therefore\,I={\tan}^{-1}{\left({e}^{x}\right)}+c$

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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
Integrate $\displaystyle \frac {1}{2}\, f\, '(x)\, w.r.t.\, x^4$, where $f(x)\, =\, tan^{-1}\, x\, +\, ln\, \sqrt{1\, +\, x}\, -\, ln\, \sqrt{1\, -\, x}$
• A. $\, -\, ln(1+x^2)\, +\, C$
• B. $\, -\, ln(1-x^2)\, +\, C$
• C. $\, -\, ln(1+x^4)\, +\, C$
• D. $\, -\, ln(1-x^4)\, +\, C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
If $\int { f\left( x \right) dx=\Psi \left( x \right) }$, then $\int { { x }^{ 5 }f\left( { x }^{ 3 } \right) } dx$ is equal to:
• A. $\dfrac { 1 }{ 3 } { x }^{ 3 }\Psi \left( { x }^{ 3 } \right) -3\int { { x }^{ 3 } } \Psi \left( { x }^{ 3 } \right) dx+C$
• B. $\dfrac { 1 }{ 4 } { x }^{ 3 }\Psi \left( { x }^{ 3 } \right) -\int { { x }^{ 2 } } \Psi \left( { x }^{ 3 } \right) dx+C$
• C. $\dfrac { 1 }{ 3 } \left[ { x }^{ 3 }\Psi \left( { x }^{ 3 } \right) -\int { { x }^{ 3 } } \Psi \left( { x }^{ 3 } \right) dx \right] +C$
• D. $\dfrac { 1 }{ 3 } { x }^{ 3 }\Psi \left( { x }^{ 3 } \right) -\int { { x }^{ 2 } } \Psi \left( { x }^{ 3 } \right) dx +C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The value of the integral $\int _ { 0 } ^ { \pi / 2 } \frac { 1 + 2 \cos x } { ( 2 + \cos x ) ^ { 2 } } d x$ is
• A. $\frac { 1 } { 4 }$
• B. $\frac { 1 } { 2 }$
• C. $\frac { -1 } { 4 }$
• D. $\frac { -1 } { 2 }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral
$\int { \cfrac { \cot { x } }{ \log { \sin { x } } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$4 \displaystyle \int \dfrac{\sqrt{a^6 + x^8}}{x} dx$ is equal to ______________________.
• A. $a^6 ln\vert \dfrac {\sqrt{a^6 + x^8} - a^3} {\sqrt {a^6 + x^8} + a^3}\vert + c$
• B. $\sqrt{a^6 + x^8} + \dfrac {a^3}{2} ln \vert \dfrac {\sqrt{a^6 + x^8} - a^3} {\sqrt {a^6 + x^8} + a^3}\vert + c$
• C. $a^6 ln \vert \dfrac {\sqrt{a^6 + x^8} + a^3} {\sqrt {a^6 + x^8} - a^3}\vert + c$
• D. $\sqrt{a^6 + x^8} + \dfrac{a^3}{2} ln \vert \dfrac {\sqrt{a^6 + x^8} + a^3} {\sqrt {a^6 + x^8} - a^3}\vert + c$