Mathematics

Evaluate $$\int { \cfrac { 1 }{ (x-1){ x }^{ 2 } }  } dx$$


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

Splitting the above expression in to partial fractions, this can be written as 

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

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

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

$$\Rightarrow \log(\dfrac{x-1}{x})+\dfrac{1}{x}$$
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Subjective Medium Published on 17th 09, 2020
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