Question

Evaluate: $\int \frac{x^3}{x+1}$ dx

Answer 1

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

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

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

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

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

$=\frac{x^3}{3}+\frac{x^2}{2}+x-\log |x+1|+c$ where $c$ is the constant of integration.