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
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