Mathematics

Solve $$\displaystyle\int \dfrac{x}{{{{\left( {x + 1} \right)}^2}}}dx$$


SOLUTION
$$\begin{array}{l} = \displaystyle\int \dfrac{x}{{{{\left( {x + 1} \right)}^2}}}dx\\ = \displaystyle\int \dfrac{{x + 1 - 1}}{{{{\left( {x + 1} \right)}^2}}}dx\\ = \displaystyle\int \left( {\dfrac{1}{{\left( {x + 1} \right)}} - \dfrac{1}{{{{\left( {x + 1} \right)}^2}}}} \right)dx\\ = \displaystyle\int \dfrac{1}{{\left( {x + 1} \right)}}dx - \displaystyle\int \dfrac{1}{{{{\left( {x + 1} \right)}^2}}}dx\\ = \ell n\left( {x + 1} \right) - \dfrac{{{{\left( {x + 1} \right)}^{ - 1}}}}{{ - 1}} + C\\ = \ell n\left( {x + 1} \right) + \dfrac{1}{{\left( {x + 1} \right)}} + C\end{array}$$
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Subjective Medium Published on 17th 09, 2020
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