Find the integral: x^ 3 -1 x^ 2 d x - Vidyadip

Question

Find the integral: $\int \frac{x^{3}-1}{x^{2}} d x$

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Answer

We have
$\int \frac{x^{3}-1}{x^{2}} d x=\int x d x-\int x^{-2} d x $
$= \left(\frac{x^{1+1}}{1+1}+C_{1}\right)-\left(\frac{x^{-2+1}}{-2+1}+C_{2}\right); $
$C_1, C_2 $ are constants of integration
$= \frac{x^{2}}{2}+C_{1}-\frac{x^{-1}}{-1}-C_{2} = \frac{x^{2}}{2}+\frac{1}{x}+C_{1}-C_{2} $
$= \frac{x^{2}}{2}+\frac{1}{x}+C $
where $C = C_1 - C_2$ is another constant of integration

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