Question

True or False:

 Consider the integral $\displaystyle{\int \frac{x^3+x}{x-1}}dx$ To evaluate the integral, we need to first perform the long division.

Answer 1

True

$\frac{x^3+x}{x-1}=x^2+\frac{x^2+x}{x-1}\\ =x^2+x+2+\frac{2}{x-1}\\ \frac{x^3+x}{x-1}=x^2-x+2-\frac{2}{x+1}\\ \mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\ =\frac{x^3}{3}-\frac{x^2}{2}+2x-2\ln \left|x+1\right|$