Mathematics

Evaluate $$\displaystyle \int {\frac{{dx}}{{x + 4 - {x^2}}}} $$


SOLUTION
$$\displaystyle \int { \dfrac { dx }{ x+4-{ x }^{ 2 } }  } =-\int { \dfrac { dx }{ { x }^{ 2 }-x-4 }  }$$
$$=\displaystyle \int { \dfrac { -dx }{ { \left( x-\dfrac { 1 }{ 2 }  \right)  }^{ 2 }-4-\dfrac { 1 }{ 4 }  }  }$$
$$=\displaystyle \int { \dfrac { -dx }{ { \left( x-\dfrac { 1 }{ 2 }  \right)  }^{ 2 }-{ \left( \dfrac { \sqrt { 17 }  }{ 2 }  \right)  }^{ 2 } }  }$$
$$=\dfrac { -1 }{ \sqrt { 17 }  } ln\left( \dfrac { x-\dfrac { 1 }{ 2 } -\dfrac { \sqrt { 17 }  }{ 2 }  }{ x-\dfrac { 1 }{ 2 } +\dfrac { \sqrt { 17 }  }{ 2 }  }  \right) +c$$
$$=\dfrac { -1 }{ \sqrt { 17 }  } ln\left( \dfrac { 2x-1-\sqrt { 17 }  }{ 2x-1+\sqrt { 17 }  }  \right) +c$$
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Subjective Medium Published on 17th 09, 2020
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