Question

Find the following integrals.
$\int \frac{x^3+5x^2-4}{x^2}dx.$

Answer 1

$\int \frac{x^3+5x^2-4}{x^2}dx=\int \frac{x^3}{x^2}dx+5\int \frac{x^2}{x^2}dx-4\int \frac{1}{x^2}dx=\int xdx+5\int 1.dx-4\int x^{-2}dx=\frac{x^2}{2}+5x-4\left(\frac{x^{-2+1}}{-2+1}\right)=\frac{x^2}{2}+5x-4\left(\frac{x^{-1}}{-1}\right)=\frac{x^2}{2}+5x+\frac{4}{x}+C.$