What is the integral of ((x^3-x^2+5x))/x ?

The integral of ((x^3-x^2+5x))/x is (x^3)/3-(x^2)/2+5x+C

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integral of ((x^3-x^2+5x))/x

\int \frac{( x ^{3} - x ^{2} + 5 x )}{x} d x

\frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 5 x + C

Solution steps

\int \frac{x ^{3} - x ^{2} + 5 x}{x} d x

Simplify

\frac{x ^{3} - x ^{2} + 5 x}{x} : x^{2} - x + 5

= \int x^{2} - x + 5 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} d x - \int x d x + \int 5 d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int x d x = \frac{x ^{2}}{2}

\int 5 d x = 5 x

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 5 x

Add a constant to the solution

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 5 x + C

What is the integral of ((x^3-x^2+5x))/x ?
The integral of ((x^3-x^2+5x))/x is (x^3)/3-(x^2)/2+5x+C