Question

$\mathrm{find \: \int \frac{x^{3}}{(x+1)^{2}}dx}$

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Answer 1

$\underline{\text{Solution :}}$

$\mathrm{Now,\:\int \frac{x^{3}}{(x+1)^{2}}dx}$

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

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

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

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

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

$\mathrm{=\frac{x^{2}}{2}+x-3x+3log|x+1|+\frac{1}{x+1}+C}$

$\mathrm{=\frac{x^{2}}{2}-2x+3log|x+1|+\frac{1}{x+1}+C}$

$\text{where C is integral constant}$

Answer 2

Answer: Solutions is attached above.........

Step-by-step explanation: