Mathematics

Integrate $$\int {({{\sin }^{ - 1}}} x{)^2}dx$$


SOLUTION
$$\begin{array}{l} I=\int { { { \left( { { { \sin   }^{ -1 } }x } \right)  }^{ 2 } }dx }  &  \\ I=\int { \left( { { { \sin   }^{ -1 } }x } \right) .\left( { { { \sin   }^{ -1 } }x } \right) dx }  & \left[ \begin{array}{l} Put\, \, { \sin ^{ -1 }  }=t \\ x=\sin  t \\ dx=\cos  tdt \end{array} \right]  \\ =\int { { t^{ 2 } }.\cos  tdt }  &  \end{array}$$
Using $$ILATE$$ formulae
$$\begin{array}{l} ={ t^{ 2 } }\sin  t-\int { 2t\sin  tdt }  \\ ={ t^{ 2 } }\sin  t-2\int { t\sin  tdt } \to Apply\, \, ILATE\, \, formulae \\ ={ t^{ 2 } }\sin  t-2\left[ { -t\cos  t+\int { \cos  tdt }  } \right]  \\ ={ t^{ 2 } }\sin  t+2t\cos  t-2\sin  t+c \\ ={ \left( { { { \sin   }^{ -1 } }x } \right) ^{ 2 } }\sin  .{ \sin ^{ -1 }  }x+2{ \sin ^{ -1 }  }x\sqrt { 1-{ x^{ 2 } } } -2\sin  .{ \sin ^{ -1 }  }x+c \\ ={ \left( { { { \sin   }^{ -1 } }x } \right) ^{ 2 } }.x+2{ \sin ^{ -1 }  }x\sqrt { 1-{ x^{ 2 } } } -2x+c \end{array}$$
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Subjective Medium Published on 17th 09, 2020
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