Mathematics

Evaluate  :  
      $$\int {\frac{{x - 5}}{{\sqrt {{x^2} + 6x + 7} }}dx} .$$


SOLUTION
$$I=\quad \int { \cfrac { x-5 }{ \sqrt { { x }^{ 2 }+6x+7 }  }  } dx=\quad \int { \cfrac { x+3-8 }{ \sqrt { { x }^{ 2 }+6x+7 }  }  } dx=\int { \cfrac { x+3 }{ \sqrt { { x }^{ 2 }+6x+7 }  }  } dx-\int { \cfrac { 8 }{ \sqrt { { x }^{ 2 }+6x+7 }  }  } dx$$
$$A=\int { \cfrac { x+3 }{ \sqrt { { x }^{ 2 }+6x+7 }  }  } dx$$
$${ x }^{ 2 }+6x+7={ t }^{ 2 }$$
$$(2x+6)dx=2tdt\Rightarrow (x+3)dx=tdt$$
$$\quad A=\int { \cfrac { tdt }{ t }  } \Rightarrow A=t+C\Rightarrow A=\sqrt { { x }^{ 2 }+6x+7 } +C$$
$$B=\int { \cfrac { 8 }{ \sqrt { { x }^{ 2 }+6x+7 }  }  } dx\quad \quad $$
$$x+3=p\Rightarrow dx=dp\quad $$
$$B=\int { \cfrac { 8dp }{ \sqrt { { p }^{ 2 }-{ (\sqrt { 2 } ) }^{ 2 } }  }  } $$
$$p=\sqrt { 2 } \sec { \theta  } \Rightarrow dp=\sqrt { 2 } \sec { \theta  } -\tan { \theta  } d\theta $$
$$B=\int { \cfrac { 8\sqrt { 2 } \sec { \theta  } -\tan { \theta  } d\theta  }{ \sqrt { 2\sec ^{ 2 }{ \theta  } -2 }  }  } \quad =\int { \cfrac { 8\sec { \theta  } -\tan { \theta  } d\theta  }{ \tan { \theta  }  } = } \int { 8 } \sec { \theta  } d\theta =8\ln { \left| \sec { \theta  } +\tan { \theta  }  \right|  } =8\ln { \left| \cfrac { p }{ \sqrt { 2 }  } +\sqrt { \cfrac { { p }^{ 2 } }{ 2 } -1 }  \right|  } =8\ln { \left| \cfrac { x+3 }{ \sqrt { 2 }  } +\sqrt { \cfrac { { (x+3) }^{ 2 }-2 }{ \sqrt { 2 }  }  }  \right|  } $$
$$\therefore$$ $$I=\sqrt { { x }^{ 2 }+6x+7 } +8\ln { \left| \cfrac { x+3+\sqrt { { x }^{ 2 }+6x+7 }  }{ \sqrt { 2 }  }  \right|  } +C$$
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Subjective Medium Published on 17th 09, 2020
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