What is the integral of (x^2-x+10)/(x^3+5x) ?

The integral of (x^2-x+10)/(x^3+5x) is 2ln|x|-1/2 ln|x^2+5|-1/(sqrt(5))arctan(x/(sqrt(5)))+C

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integral of (x^2-x+10)/(x^3+5x)

\int \frac{x ^{2} - x + 10}{x ^{3} + 5 x} d x

2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5}) + C

Solution steps

\int \frac{x ^{2} - x + 10}{x ^{3} + 5 x} d x

Take the partial fraction of

\frac{x ^{2} - x + 10}{x ^{3} + 5 x} : \frac{2}{x} + \frac{- x - 1}{x ^{2} + 5}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{- x - 1}{x ^{2} + 5} d x = - \frac{1}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5})

= 2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5})

Add a constant to the solution

= 2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5}) + C

\int \frac{2 x ^{4}}{x ^{3} - x ^{2} + x - 1} d x

\int \frac{1 - 25 x ^{2}}{x ^{2}} d x

\int 3 \frac{cos ( x ) + sin ( x )}{sin ( 2 x )} d x

e^{2 x + 1}

f (x) = 5 x^{- 2}

What is the integral of (x^2-x+10)/(x^3+5x) ?
The integral of (x^2-x+10)/(x^3+5x) is 2ln|x|-1/2 ln|x^2+5|-1/(sqrt(5))arctan(x/(sqrt(5)))+C