Question

Evaluate; $\int \left(\frac{3x^4-5x^3+4x^2-x+2}{x^3}\right)dx$

Answer 1

$\begin{array}{c}\int \left(\frac{3x^4-5x^3+4x^2-x+2}{x^3}\right)dx=\int \left(3x-5+\frac{4}{x}-\frac{1}{x^2}+\frac{2}{x^3}\right)dx\left[divideeachtermby{x}^3\right]\\ =3\int xdx-5\int dx+4\int \frac{1}{4}dx-\int x^{-2}dx+2\int x^{-3}dx\\ =3.\frac{x^2}{2}-5x+4\log \left|x\right|-\left(-\frac{1}{x}\right)+2\left(\frac{x^{-2}}{-2}\right)+C\\ =\frac{3x^2}{2}-5x+4\log \left|x\right|-\left(-\frac{1}{x}\right)+2\left(\frac{x^{-2}}{-2}\right)+C\\ =\frac{3x^2}{2}-5x+4\log \left|x\right|+\frac{1}{x}-\frac{1}{x^2}+C.\end{array}$