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

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

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

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

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

Solution steps

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

Simplify

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

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

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

\int 5 x d x = \frac{5 x ^{2}}{2}

\int 2 d x = 2 x

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

Add a constant to the solution

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

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