$\int \frac{3 x-2}{(x+1)(x-2)^2} d x=$ (where $C$ is a constant of integration.)

$\int \frac{3 x-2}{(x+1)(x-2)^2} d x=$ (where $C$ is a constant of integration.)
  1. $\frac{-5}{9} \log (x+1)+\frac{5}{9} \log (x-2)-\frac{4}{3} \times \frac{1}{(x-2)}+C$
  2. $\frac{1}{9} \log (x+1)+\frac{5}{9} \log (x-2)-\frac{4}{3} \times \frac{1}{(x-2)}+C$
  3. $\frac{-5}{9} \log (x+1)+\frac{5}{9} \log (x-2)+\frac{4}{3} \times \frac{1}{x-2}+C$
  4. $\frac{-5}{9} \log (x+1)+\frac{1}{9} \log (x-2)+\frac{1}{x-2}+C$

Solution

$\int \frac{3 x-2}{(x+1)(x-2)^2} d x=\int\left\{\frac{-5}{9(x+1)}+\frac{5}{9(x-2)}+\frac{4}{3(x-2)^2}\right\} d x$ [Using partial fraction] $=-\frac{5}{9} \log |x+1|+\frac{5}{9} \log |x-2|-\frac{4}{3(x-2)}+C$

Asked in: MHT CET 2022 (06 Aug Shift 2)

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