Mathematics

# The value of $\displaystyle \int_{0}^{2\pi}\frac{dx}{e^{\sin x}+1}$

$\displaystyle \pi$

##### SOLUTION
Let $\displaystyle I=\int_{0}^{2\pi}\frac{dx}{e^{\sin x}+1}$...(1)

$\displaystyle I=\int _{ 0 }^{ 2\pi } \frac { dx }{ e^{ -\sin x }+1 }$

$\displaystyle I=\int _{ 0 }^{ 2\pi } \frac { e^{ \sin x } }{ e^{ \sin x }+1 }$...(2)

$\displaystyle 2I=\int _{ 0 }^{ 2\pi } \frac { e^{ \sin x }+1 }{ e^{ \sin x }+1 } dx$

$\displaystyle 2I=2 \pi$

$\Rightarrow I=\pi$

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

#### Realted Questions

Q1 Single Correct Hard
If $\displaystyle I = \int_{0}^{\infty}\frac{\sqrt{x}\:d\:x}{(1+x)(2+x)(3+x)}$ then $I$ equals
• A. $\displaystyle \frac{\pi}{2}(2\sqrt{2}+\sqrt{3}-1)$
• B. $\displaystyle \frac{\pi}{2}(2\sqrt{2}-\sqrt{3}+1)$
• C. None of these
• D. $\displaystyle \frac{\pi}{2}(2\sqrt{2}-\sqrt{3}-1)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
If $f\left( \cfrac { 3x-4 }{ 3x+4 } \right) =x+2$, then $\int { f(x) } dx$ is
• A. ${ e }^{ x+2 }\log { \left| \cfrac { 3x-4 }{ 3x+4 } \right| } +c$
• B. $\cfrac { 8 }{ 3 } \log { \left| 1-x \right| } +\cfrac { x }{ 3 } +c$
• C. ${ e }^{ \left[ \left( 3x-4 \right) /\left( 3x+4 \right) \right] }-\cfrac { { x }^{ 2 } }{ 2 } -2x+c$
• D. $-\cfrac { 8 }{ 3 } \log { \left| 1-x \right| } +\cfrac { 2 }{ 3 } x+c\quad$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\displaystyle \int\cfrac{dx}{x\left(x^{4}-1\right)}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Integrate $\int x.\cos^{-1}x$

$\displaystyle \int \dfrac {4x+5}{2x^2+5x+18}dx$