Mark the correct alternative in each of the following:

Question:

Mark the correct alternative in each of the following:

$\int \frac{x^{3}}{x+1} d x$

A. $x+\frac{x^{2}}{2}+\frac{x^{3}}{3}-\log |1-x|+C$

B. $x+\frac{x^{2}}{2}-\frac{x^{3}}{3}-\log |1-x|+C$

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

D. $x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\log |1+x|+C$

Solution:

$=\int \frac{x^{3}+1}{x+1} d x-\int \frac{1}{x+1} d x$

$=\int \frac{(x+1)\left(x^{2}-x+1\right)}{x+1} d x-\int \frac{1}{x+1} d x$

$=\int\left(x^{2}-x+1\right) d x-\int \frac{1}{x+1} d x$

$=\frac{x^{3}}{3}-\frac{x^{2}}{2}+x-\log (1+x)+c$

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