What is the integral of (3-5x+7x^2+x^3)/(x^2) ?

The integral of (3-5x+7x^2+x^3)/(x^2) is -3/x-5ln|x|+7x+(x^2)/2+C

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integral of (3-5x+7x^2+x^3)/(x^2)

\int \frac{3 - 5 x + 7 x ^{2} + x ^{3}}{x ^{2}} d x

- \frac{3}{x} - 5 ln ∣ x ∣ + 7 x + \frac{x ^{2}}{2} + C

Solution steps

\int \frac{3 - 5 x + 7 x ^{2} + x ^{3}}{x ^{2}} d x

Expand

\frac{3 - 5 x + 7 x ^{2} + x ^{3}}{x ^{2}} : \frac{3}{x ^{2}} - \frac{5}{x} + 7 + x

= \int \frac{3}{x ^{2}} - \frac{5}{x} + 7 + 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{3}{x ^{2}} d x - \int \frac{5}{x} d x + \int 7 d x + \int x d x

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

\int 7 d x = 7 x

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

= - \frac{3}{x} - 5 ln ∣ x ∣ + 7 x + \frac{x ^{2}}{2}

Add a constant to the solution

= - \frac{3}{x} - 5 ln ∣ x ∣ + 7 x + \frac{x ^{2}}{2} + C

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What is the integral of (3-5x+7x^2+x^3)/(x^2) ?
The integral of (3-5x+7x^2+x^3)/(x^2) is -3/x-5ln|x|+7x+(x^2)/2+C