Mathematics

Evaluate:
$$\displaystyle \int \dfrac{1}{(2x - 7)} \sqrt{(x - 3)(x - 4)}\ dx$$


SOLUTION
$$I=\displaystyle \int { \dfrac { dx }{ \left( 2x-7 \right)  } \sqrt { \left( x-3 \right) \left( x-4 \right)  }  } $$

$$=\displaystyle \int { \dfrac { \sqrt { \left( x-3 \right) \left( x-4 \right)  }  }{ \left( 2x-7 \right)  } dx } $$

$$=\displaystyle \int { \dfrac { \sqrt { { x }^{ 2 }-3x-4x+12 }  }{ 2x-7 } dx } $$

$$=\displaystyle \int { \dfrac { \sqrt { { x }^{ 2 }-7x+12 }  }{ 2x-7 } dx } $$

$$=\displaystyle \int { \dfrac { \sqrt { { x }^{ 2 }-2.\dfrac { 7 }{ 2 } x+{ \left( \dfrac { 7 }{ 2 }  \right)  }^{ 2 }-{ \left( \dfrac { 7 }{ 2 }  \right)  }^{ 2 }+12 }  }{ 2x-7 } dx } $$

$$=\displaystyle \int { \dfrac { \sqrt { { \left( x-\dfrac { 7 }{ 2 }  \right)  }^{ 2 }+12-\dfrac { 49 }{ 4 }  }  }{ 2x-7 } dx } $$

$$=\displaystyle \int { \dfrac { \sqrt { { \left( x-\dfrac { 7 }{ 2 }  \right)  }^{ 2 }-{ \left( \dfrac { 1 }{ 2 }  \right)  }^{ 2 } }  }{ 2\left( 1-\dfrac { 7 }{ 2 }  \right)  } dx } $$

Let $${ \left( x-\dfrac { 7 }{ 2 }  \right)  }^{ 2 }-{ \left( \dfrac { 1 }{ 2 }  \right)  }^{ 2 }=t^2$$
$$\Rightarrow 2\left( x-\dfrac { 7 }{ 2 }  \right) dx=2t\;dt$$ $$\Rightarrow dx=\dfrac { 2tdt }{ 2\left( x-\dfrac { 7 }{ 2 }  \right)  } $$

$$\Rightarrow I=\displaystyle \int { \dfrac { t.tdt }{ 2{ \left( x-\dfrac { 7 }{ 2 }  \right)  }^{ 2 } }  } $$

$$=\displaystyle \dfrac { 1 }{ 4 } \int { \dfrac { t^{ 2 }dt }{ { \left( x-\dfrac { 7 }{ 2 }  \right)  }^{ 2 } }  } $$

$$=\displaystyle \dfrac { 1 }{ 4 } \int { \dfrac { t^{ 2 }+\dfrac { 1 }{ 4 } -\dfrac { 1 }{ 4 } dt }{ { t }^{ 2 }+\dfrac { 1 }{ 4 }  }  } $$

$$=\displaystyle \dfrac { 1 }{ 4 } \int { 1dt-\dfrac { 1 }{ 4 } \int { \dfrac { 1 }{ t^{ 2 }+\dfrac { 1 }{ 4 }  } dt }  } $$

$$=\displaystyle \dfrac { 1 }{ 4 } t-\dfrac { 1 }{ 4\times \dfrac { 1 }{ 4 }  } \tan ^{ -1 }{ \dfrac { t }{ \dfrac { 1 }{ 4 }  } +c } $$

$$=\dfrac { 1 }{ 4 } t-\tan ^{ -1 }{ \dfrac { t }{ \dfrac { 1 }{ 4 }  } +c } $$
$$=\dfrac { 1 }{ 4 } \sqrt { { x }^{ 2 }-7x+12 } -\tan ^{ -1 }{ 4\sqrt { \left( x-3 \right) \left( x-4 \right)  } +c } $$
Hence, the answer is $$\dfrac { 1 }{ 4 } \sqrt { { x }^{ 2 }-7x+12 } -\tan ^{ -1 }{ 4\sqrt { \left( x-3 \right) \left( x-4 \right)  } +c } .$$
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Subjective Hard Published on 17th 09, 2020
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