Question

Integrate:

$\int \frac{x^2+1}{{(x+1)}^3(x-2)}dx$

Answer 1

Consider the given integral.

$I=\int \frac{x^2+1}{{(x+1)}^3(x-2)}dx$

$I=\frac{-5}{27}\int \frac{dx}{(x+1)}+\frac{4}{9}\int \frac{dx}{{(x+1)}^2}-\frac{2}{3}\int \frac{dx}{{(x+1)}^3}+\frac{5}{27}\int \frac{dx}{(x-2)}$

$I=-\frac{5}{27}\ln (x+1)+\frac{4}{9}(-\frac{1}{(x+1)})-\frac{2}{3}(-\frac{1}{2{(x+2)}^2})+\frac{5}{27}\ln (x-2)+C$

$I=\frac{5}{27}(\ln (x-2)-\ln (x+1))-\frac{4}{9}\frac{1}{(x+1)}+\frac{1}{3{(x+2)}^2}+C$

$I=\frac{5}{27}\ln \left(\frac{x-2}{x+1}\right)-\frac{4}{9}\frac{1}{(x+1)}+\frac{1}{3{(x+2)}^2}+C$

Hence, this is the answer.