What is the integral of (x^3+5x^2-x+4)/x ?

The integral of (x^3+5x^2-x+4)/x is (x^3)/3+(5x^2)/2-x+4ln|x|+C

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integral of (x^3+5x^2-x+4)/x

\int \frac{x ^{3} + 5 x ^{2} - x + 4}{x} d x

\frac{x ^{3}}{3} + \frac{5 x ^{2}}{2} - x + 4 ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{3} + 5 x ^{2} - x + 4}{x} d x

Expand

\frac{x ^{3} + 5 x ^{2} - x + 4}{x} : x^{2} + 5 x - 1 + \frac{4}{x}

= \int x^{2} + 5 x - 1 + \frac{4}{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 5 x d x - \int 1 d x + \int \frac{4}{x} d x

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

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

\int 1 d x = x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

= \frac{x ^{3}}{3} + \frac{5 x ^{2}}{2} - x + 4 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{5 x ^{2}}{2} - x + 4 ln ∣ x ∣ + C

What is the integral of (x^3+5x^2-x+4)/x ?
The integral of (x^3+5x^2-x+4)/x is (x^3)/3+(5x^2)/2-x+4ln|x|+C