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:
$=\int \frac{x^{3}+1}{x+1} d x-\int \frac{1}{x+1} d x$
$=\int \frac{(x+1)\left(x^{2}-x+1\right)}{x+1} d x-\int \frac{1}{x+1} d x$
$=\int\left(x^{2}-x+1\right) d x-\int \frac{1}{x+1} d x$
$=\frac{x^{3}}{3}-\frac{x^{2}}{2}+x-\log (1+x)+c$