Mathematics

# Integrate $\displaystyle \int\dfrac{e^{x}-1}{e^{x}+1}dx$

$2\log(e^{-x}+1)-x+C$

##### SOLUTION
$\int_{}^{} {\dfrac{{{e^x} - 1}}{{{e^x} + 1}}dx}$
putting ${e^x} + 1 = t$
$\Rightarrow {e^x}dx = dt$
$\Rightarrow dx = \dfrac{{dt}}{{{e^x}}}$
$\Rightarrow dx = \dfrac{{dt}}{{t - 1}}$
$= \int_{}^{} {\dfrac{{t - 2}}{{t\left( {t - 1} \right)}}dt}$
$= \int_{}^{} {\left( {\dfrac{2}{t} - \dfrac{1}{{t - 1}}} \right)dt}$
$= 2\log t - \log \left( {t - 1} \right) + c$
$= 2\log \left( {{e^x} + 1} \right) - \log \left( {{e^x}} \right) + c$
$= 2\log \left( {{e^x} + 1} \right) - x + c$

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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
If $\displaystyle\int\dfrac{8}{{(4n - 1)\;(4n + 3)}}dx$ .find integration of this function.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\int\limits_0^\infty {\dfrac{{dx}}{{({x^2} + {a^2})({x^2} + {b^2})}}} =$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate: $\displaystyle \int \frac{\log \left [ x+\sqrt{1+x^{2}} \right ]}{\sqrt{1+x^{2}}}dx$
• A. $\displaystyle \left [ \log \left ( x+\sqrt{1+x^{2}} \right ) \right ]^{2}$
• B. $\displaystyle 2\left [ \log \left ( x+\sqrt{1+x^{2}} \right ) \right ]^{2}$
• C. $\displaystyle \frac{1}{2}\left [ \log \left ( x+\sqrt{1+x^{2}} \right ) \right ]$
• D. $\displaystyle \frac{1}{2}\left [ \log \left ( x+\sqrt{1+x^{2}} \right ) \right ]^{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Evaluate the following integrals:

$\displaystyle \int \dfrac{x^3}{2x + 1}dx$
• A. $\dfrac{x^3}{3}+\dfrac{x^2}{8}+\dfrac{1}{8x}-\dfrac{1}{16}log(2x+1)+C$
• B. $\dfrac{x^3}{6}-\dfrac{x^2}{6}-\dfrac{1}{8x}-\dfrac{1}{16}log(2x+1)+C$
• C. $\dfrac{x^3}{3}-\dfrac{x^2}{8}-\dfrac{1}{8x}-\dfrac{1}{16}log(2x+1)+C$
• D. $\dfrac{x^3}{6}-\dfrac{x^2}{8}+\dfrac{1}{8x}-\dfrac{1}{16}log(2x+1)+C$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$