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

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

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

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

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

Solution steps

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

Expand

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

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

Add a constant to the solution

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

\int_{- 2}^{1} 2 - x - x^{2} d x

n = 0 \sum \infty \frac{n}{( n - 2 ) !}

\frac{d}{d x} (\frac{x ^{2} - 3 x + 8}{x ^{2} + 9})

\int 12 + 4 x^{2} d x

\frac{d}{d x} ((\frac{1 + sin ( x )}{1 - cos ( x )})^{2})

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