Mathematics

# Prove that:$\int _{ }^{ }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } dx=\cfrac { x }{ 2 } \sqrt { { a }^{ 2 }-{ x }^{ 2 } } +\cfrac { { a }^{ 2 } }{ 2 } \sin ^{ -1 }{ \left( \cfrac { x }{ a } \right) } +c$

##### SOLUTION
$I=\displaystyle \int \sqrt {a^2-x^2}dx-----(1)$
$I=\left[ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } \displaystyle \int { 1-dx } \displaystyle \int { \dfrac { d }{ dx } } \sqrt { { a }^{ 2 }-{ x }^{ 2 } } \left( \displaystyle \int { 1dx } \right) \right] dx$
$I=x\sqrt { { a }^{ 2 }-{ x }^{ 2 } } -\displaystyle \int { \dfrac { -{ x }^{ 2 } }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } dx }$
$I=x\sqrt { { a }^{ 2 }-{ x }^{ 2 } } -\displaystyle \int { \dfrac { -{ a }^{ 2 }+{ a }^{ 2 }-{ x }^{ 2 } }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } } dx$
$I=x\sqrt { { a }^{ 2 }-{ x }^{ 2 } } -\displaystyle \int { \dfrac { -{ a }^{ 2 } }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } } dx-\displaystyle \int { \dfrac { { a }^{ 2 }-{ x }^{ 2 } }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } } dx$
$I=x\sqrt { { a }^{ 2 }-{ x }^{ 2 } } +{ a }^{ 2 }\displaystyle \int { \dfrac { 1 }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } } dx-\displaystyle \int { \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } dx$
$I=x\sqrt { { a }^{ 2 }-{ x }^{ 2 } } +{ a }^{ 2 }\sin ^{ -2 }{ \left( \dfrac { x }{ a } \right) } -I+C$
$2I=x\sqrt { { a }^{ 2 }-{ x }^{ 2 } } +{ a }^{ 2 }\sin ^{ -2 }{ \left( \dfrac { x }{ a } \right) } +C$
$I=\dfrac { x }{ 2 } \sqrt { { a }^{ 2 }-{ x }^{ 2 } } +\dfrac { { a }^{ 2 } }{ z } \sin ^{ -2 }{ \left( \dfrac { x }{ a } \right) } +C$

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

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