Mathematics

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


ANSWER

$$A+B=0$$


SOLUTION
$$\int{\dfrac{dx}{(x+1)(x-2)}}$$

This can be written as 

$$\Rightarrow \int{(\dfrac{-1}{3}\dfrac{1}{x+1}+\dfrac{1}{3}\dfrac{1}{x-2})}dx$$

$$\Rightarrow \dfrac{-1}{3}\int {\dfrac{1}{x+1}dx +\dfrac{1}{3}\int\dfrac{1}{x-2}}dx$$

$$\Rightarrow \dfrac{-1}{3}\log(x+1)+\dfrac{1}{3}\log(x-2)+C$$

Thus comparing this whith given expression,

$$\Rightarrow A=\dfrac{-1}{3}, B=\dfrac{1}{3}$$

$$\Rightarrow A+B=0$$
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Single Correct Medium Published on 17th 09, 2020
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