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

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

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

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

$$I=\displaystyle\int{dt}$$

$$I=t+c$$

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

View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

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. $$\, -\, ln(1-x^4)\, +\, C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$\dfrac { 1 }{ 3 } { x }^{ 3 }\Psi \left( { x }^{ 3 } \right) -\int { { x }^{ 2 } } \Psi \left( { x }^{ 3 } \right) dx  +C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$\frac { -1 } { 2 }$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$\sqrt{a^6 + x^8} + \dfrac{a^3}{2} ln \vert \dfrac {\sqrt{a^6 + x^8} + a^3} {\sqrt {a^6 + x^8} - a^3}\vert + c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer