Mathematics

# $\displaystyle \int_{\pi /4}^{3\pi /4}\frac{dx}{1+\cos x}$ is equal to

##### ANSWER

$2$

##### SOLUTION
Let $\displaystyle I=\int { \dfrac { 1 }{ 1+\cos { x } } } dx$

Substitute $\displaystyle t=\tan { \dfrac { x }{ 2 } } \Rightarrow dt=\dfrac { 1 }{ 2 } dxsec^{ 2 }x$

$\displaystyle I=\int { \dfrac { 2 }{ \left( { u }^{ 2 }+1 \right) \left( \dfrac { 1-{ u }^{ 2 } }{ { u }^{ 2 }+1 } +1 \right) } } du=\int { du } =u=\tan { \dfrac { x }{ 2 } }$

$\displaystyle \therefore \int _{ \dfrac { \pi }{ 4 } }^{ \dfrac { 3\pi }{ 4 } }{ \dfrac { dx }{ 1+\cos { x } } } =\left[ \tan { \dfrac { x }{ 2 } } \right] _{ \dfrac { \pi }{ 4 } }^{ \dfrac { 3\pi }{ 4 } }=2$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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#### Realted Questions

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Q5 Passage Hard
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1 Verified Answer | Published on 17th 09, 2020