Mathematics

Integrate:
$$\int {\dfrac{{{x^2} + 1}}{{{{(x + 1)}^3}(x - 2)}}} dx$$


SOLUTION

Consider the given integral.

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

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

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

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

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

 

Hence, this is the answer.

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