What is the integral of (x^2-x+8)/(x^3+2x) ?

The integral of (x^2-x+8)/(x^3+2x) is 4ln|x|-3/2 ln|x^2+2|-1/(sqrt(2))arctan(x/(sqrt(2)))+C

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integral of (x^2-x+8)/(x^3+2x)

\int \frac{x ^{2} - x + 8}{x ^{3} + 2 x} d x

4 ln ∣ x ∣ - \frac{3}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2}) + C

Solution steps

\int \frac{x ^{2} - x + 8}{x ^{3} + 2 x} d x

Take the partial fraction of

\frac{x ^{2} - x + 8}{x ^{3} + 2 x} : \frac{4}{x} + \frac{- 3 x - 1}{x ^{2} + 2}

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int \frac{- 3 x - 1}{x ^{2} + 2} d x = - \frac{3}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2})

= 4 ln ∣ x ∣ - \frac{3}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2})

Add a constant to the solution

= 4 ln ∣ x ∣ - \frac{3}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2}) + C

\int (x + 1) (3 x + 2) d x

f (x) = x^{3} - 12 x

\int (\frac{3}{5}) cos (x) d x

\frac{d}{d x} (\frac{2 x + 1}{x + 2})

\frac{d}{d x} (ln (\frac{7 + e ^{x}}{7 - e ^{x}}))

What is the integral of (x^2-x+8)/(x^3+2x) ?
The integral of (x^2-x+8)/(x^3+2x) is 4ln|x|-3/2 ln|x^2+2|-1/(sqrt(2))arctan(x/(sqrt(2)))+C