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

6 t (\frac{d y}{d t}) + 5 y = t

\frac{\partial}{\partial w} (\frac{w}{h ( w ) ^{2}})

y = \frac{1}{6 x - 1}

\int_{0}^{2} (x) d x

\int e^{2 θ} + e^{2 θ} d θ

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