Mathematics

$$\displaystyle \int \frac{dx}{\left ( x+1 \right )\left ( x-2 \right )}=A\log \left ( x+1 \right )+B\log \left ( x-2 \right )+C,$$ where


ANSWER

$$\displaystyle A+B=0$$

$$\displaystyle A : B=-1$$


SOLUTION

$$ \displaystyle \int  \frac { dx }{ \left( x+1 \right) \left( x-2 \right)  } $$


$$  \displaystyle =\frac { 1 }{ 3 } \int { \frac { \left( x+1 \right) -\left( x-2 \right)  }{ \left( x+1 \right) \left( x-2 \right)  }  } dx$$

$$ \displaystyle =\frac { 1 }{ 3 } \left[ \int { \frac { 1 }{ x-2 } dx } -\int { \frac { 1 }{ x+1 } dx }  \right] $$

$$ \displaystyle =\frac { 1 }{ 3 } \log { \left( x-2 \right)  } -\frac { 1 }{ 3 } \log { \left( x+1 \right)  } +C$$

Hence, we get $$A = -\dfrac13$$ and $$B = \dfrac13$$

$$\therefore A+B = 0$$ and $$A:B = -1$$

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Multiple Correct Medium Published on 17th 09, 2020
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