Question

Evaluate: $\int \frac{x^3}{x+2}dx$

Answer 1

$\int \frac{x^{3}dx}{x+2}=\int \frac{(x^3+8-8)dx}{x+2}$

$=\int \frac{(x^3+8)dx}{(x+2)}-\int \frac{8dx}{x+2}$

$=\int \frac{(x+2)(x^2-2x+4)dx}{(x+2)}-8\int \frac{dx}{x+2}$

$=\int (x^2-2x+4)dx-8\int \frac{dx}{x+2}$

$=\frac{x^3}{3}-\frac{2x^2}{2}+4x-8log|x+2|+c$

$=\frac{x^3}{3}-x^2+4x-8log|x+2|+c$

Where c is the constant of integration