Mathematics

Evaluate: $$\int {\dfrac{{dx}}{{\left( {5 - 8x - {x^2}} \right)}}dx} $$


SOLUTION
$$\begin{array}{l} We\, have \\ \int { \dfrac { { dx } }{ { \left( { 5-8x-{ x^{ 2 } } } \right)  } } dx } =-\int { \dfrac { { dx } }{ { \left( { { x^{ 2 } }+8x-5 } \right)  } }  }  \\ =-\int { \dfrac { { dx } }{ { \left\{ { \left( { { x^{ 2 } }+8x+16 } \right) -21 } \right\}  } }  }  \\ =-\int { \dfrac { { dx } }{ { \left\{ { { { \left( { x+4 } \right)  }^{ 2 } }-{ { \left( { \sqrt { 21 }  } \right)  }^{ 2 } } } \right\}  } }  }  \\ =\int { \dfrac { { dx } }{ { \left\{ { { { \left( { \sqrt { 21 }  } \right)  }^{ 2 } }-{ { \left( { x+4 } \right)  }^{ 2 } } } \right\}  } }  }  \\ =\int { \dfrac { { dt } }{ { \left\{ { { { \left( { \sqrt { 21 }  } \right)  }^{ 2 } }-{ t^{ 2 } } } \right\}  } } \, \, \, \, \, where\, \, \left( { x+4 } \right) =t }  \\ =\dfrac { 1 }{ { 2\sqrt { 21 }  } } .\log  \left| { \dfrac { { \sqrt { 21 } +t } }{ { \sqrt { 21 } -t } }  } \right| +C \\ =\dfrac { 1 }{ { 2\sqrt { 21 }  } } .\log  \left| { \dfrac { { \sqrt { 21 } +4+x } }{ { \sqrt { 21 } -4-x } }  } \right| +C.  \end{array}$$

Hence, this is the required answer.
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Subjective Medium Published on 17th 09, 2020
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