Mathematics

Evaluate:
$$\displaystyle \int \dfrac { x ^ { 3 } } { ( x + 1 ) ^ { 2 } } d x$$


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

$$I=\displaystyle\int \frac{x^{2}}{(x+1)^{2}}xdx$$

$$u= x+1$$

$$\Rightarrow x=u-1, dx=du$$

$$I=\displaystyle\int \frac{(u-1)^{2}(u-1)du}{u^{2}}$$

$$=\displaystyle\int \frac{(u-1)^{3}}{u^{2}}du$$

$$=\displaystyle\int \frac{u^{3}-3u^{2}+3u-1}{u^{2}}.du$$

$$=\displaystyle\int u-3+\frac{3}{u}-\frac{1}{u^{2}}du$$

$$=\dfrac{u^{2}}{2}-3u+3ln\left | u \right |+\dfrac{1}{u}+c$$

$$=\dfrac{(x+1)^{2}}{2}-3(x+1)+3ln\left | x+1 \right |+\dfrac{1}{x+1}+c$$
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Subjective Medium Published on 17th 09, 2020
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