Mathematics

Find the integral of $$\sqrt { { x }^{ 2 }-{ a }^{ 2 } } $$ w.r.t $$x$$ and hence evaluate $$\int { \sqrt { { x }^{ 2 }-8x+7 }  } dx\quad $$


SOLUTION
$$I=\int { \sqrt { { x }^{ 2 }-{ a }^{ 2 } }  } dx=\sqrt { { x }^{ 2 }-{ a }^{ 2 } } .x-\cfrac { 1 }{ 2 } { \left( { x }^{ 2 }-{ a }^{ 2 } \right)  }^{ 1/2-1 }(2x)$$
$$=x\sqrt { { x }^{ 2 }-{ a }^{ 2 } } -\int { \sqrt { { x }^{ 2 }-{ a }^{ 2 } }  } -{ a }^{ 2 }\int { \cfrac { 1 }{ \sqrt { { x }^{ 2 }-{ a }^{ 2 } }  }  } $$
$$\quad I=x\sqrt { { x }^{ 2 }-{ a }^{ 2 } } -1-{ a }^{ 2 }\log { \left| x+\sqrt { { x }^{ 2 }-{ a }^{ 2 } }  \right|  } \quad $$
$$I=\int { \sqrt { { x }^{ 2 }-8x+7 }  } dx=\int { \sqrt { { \left( x-4 \right)  }^{ 2 }-{ 3 }^{ 2 } }  } dx=\cfrac { x-4 }{ 2 } \sqrt { { \left( x-4 \right)  }^{ 2 }-{ 3 }^{ 2 } } -\cfrac { 9 }{ 2 } \log { \left| \left( x-4 \right) \sqrt { { \left( x-4 \right)  }^{ 2 }-{ 3 }^{ 2 } }  \right|  } +c\quad \quad $$
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Subjective Medium Published on 17th 09, 2020
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