Mathematics

$$\displaystyle \int \frac{1}{e^{x}+1}dx.$$


ANSWER

$$\displaystyle x-\log \left ( e^{x} +1\right ).$$


SOLUTION
Let $$ \displaystyle I=\int  \frac { 1 }{ e^{ x }+1 } dx=\int  \frac { e^{ x }dx }{ e^{ x }\left( e^{ x }+1 \right)  } $$
Put $$ \displaystyle e^{ x }=t\Rightarrow e^{ x }dx=dt$$
$$ \displaystyle I=\int  \frac { dt }{ t\left( t+1 \right)  } =\int  \left( \frac { 1 }{ t } -\frac { 1 }{ t+1 }  \right) dt$$
$$ \displaystyle =\log  t-\log  \left( t+1 \right) =\log  e^{ x }-\log  \left( e^{ x }+1 \right) $$
$$ \displaystyle =x-\log  \left( e^{ x }+1 \right) $$
View Full Answer

Its FREE, you're just one step away


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

Realted Questions

Q1 Subjective Medium
Solve  $$\int _ { 0 } ^ { 100 } \left[ \tan ^ { - 1 } ( x ) \right] d x $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Integrate by using suitable substitution
(1). $$\int {\sqrt {3 - 2s} } $$ ds
(2). $$\int {\csc \left( {{{v - \pi } \over 2}} \right)} \cot \left( {{{v - \pi } \over 2}} \right)dv$$
(3). $$\int\limits_\pi ^{2\pi } {\theta d\theta } $$
(4). $$\int\limits_0^{\sqrt x } {x\sin {x^2}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 One Word Medium
$$\displaystyle \int_{0}^{\infty }\left ( \cot ^{-1}x \right )^{2}dx= \frac{\pi}{k} \log 2$$. Find the value of $$k$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
$$\int { \cos ^{ 4 }{ 2x }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Solve $$\displaystyle\int {\dfrac{x}{{\sqrt {4 - {x^2}} }}} dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer