Mathematics

# Find the value of $\displaystyle I=\int _{ 0 }^{ \pi /2 }{ \frac { \sqrt { \sin { x } } }{ \sqrt { \sin { x } } +\sqrt { \cos { x } } } dx }$

$\displaystyle \frac { \pi }{ 4 }$

##### SOLUTION
$\displaystyle I=\int _{ 0 }^{ \pi /2 }{ \frac { \sqrt { \sin { x } } }{ \sqrt { \sin { x } } +\sqrt { \cos { x } } } dx } =\int _{ 0 }^{ \pi /2 }{ \frac { \sqrt { \sin { \left( \frac { \pi }{ 2 } -x \right) } } }{ \sqrt { \sin { \left( \frac { \pi }{ 2 } -x \right) } } +\sqrt { \cos { \left( \frac { \pi }{ 2 } -x \right) } } } dx }$   ....(1)

Using $\displaystyle \int _{ a }^{ b }{ f\left( x \right)dx } =\int _{ a }^{ b }{ f\left( a+b-x \right)dx }$

$\displaystyle \Rightarrow I=\int _{ 0 }^{ \pi /2 }{ \frac { \sqrt { \cos { x } } }{ \sqrt { \cos { x } } +\sqrt { \sin { x } } } dx }$   ...(2)

$\displaystyle 2I=\int _{ 0 }^{ \pi /2 }{ \left( \frac { \sqrt { \sin { x } } }{ \sqrt { \sin { x } } +\sqrt { \cos { x } } } +\frac { \sqrt { \cos { x } } }{ \sqrt { \cos { x } } +\sqrt { \sin { x } } } \right) } dx$

$\displaystyle =\int _{ 0 }^{ \pi /2 }{ \left( \frac { \sqrt { \sin { x } } +\sqrt { \cos { x } } }{ \sqrt { \sin { x } } +\sqrt { \cos { x } } } \right) dx } =\int _{ 0 }^{ \pi /2 }{ dx } ={ \left[ x \right] }_{ 0 }^{ \pi /2 }=\frac { \pi }{ 2 }$

$\displaystyle \therefore I=\frac { \pi }{ 4 }$

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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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