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

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

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

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

\frac{x ^{3}}{3} + x^{2} + 5 ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{x ^{3}}{3} + x^{2} + 5 ln ∣ x ∣

Add a constant to the solution

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

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