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

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

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

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

ln ∣ x ∣ + 3 x + \frac{x ^{2}}{2} + C

Solution steps

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

Simplify

\frac{x + 3 x ^{2} + x ^{3}}{x ^{2}} : \frac{1 + 3 x + x ^{2}}{x}

= \int \frac{1 + 3 x + x ^{2}}{x} d x

Expand

\frac{1 + 3 x + x ^{2}}{x} : \frac{1}{x} + 3 + x

= \int \frac{1}{x} + 3 + 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 \frac{1}{x} d x + \int 3 d x + \int x d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int 3 d x = 3 x

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

= ln ∣ x ∣ + 3 x + \frac{x ^{2}}{2}

Add a constant to the solution

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

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