Mathematics

# The value of $\displaystyle \int_{-\pi /2}^{\pi /2}\frac{dx}{\sin \:^{3}x+\sin x}$is

$0$

##### SOLUTION
Let $\displaystyle I=\int _{ -\dfrac { \pi }{ 2 } }^{ \dfrac { \pi }{ 2 } }{ \dfrac { dx }{ \sin ^{ 3 }{ x } +\sin { x } } }$
Using $\int _{ a }^{ b }{ f\left( x \right) } dx=\int _{ a }^{ b }{ f\left( a+b-x \right) } dx$
$\displaystyle I=\int _{ -\dfrac { \pi }{ 2 } }^{ \dfrac { \pi }{ 2 } }{ \dfrac { dx }{ \sin ^{ 3 }{ \left( -x \right) } +\sin { \left( -x \right) } } } dx$
$\displaystyle =-\int _{ -\dfrac { \pi }{ 2 } }^{ \dfrac { \pi }{ 2 } }{ \dfrac { dx }{ \sin ^{ 3 }{ x } +\sin { x } } } dx=-I\\ \therefore 2I=I\Rightarrow I=0$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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