What is the integral of (x^2-x+12)/(x^3+3x) ?

The integral of (x^2-x+12)/(x^3+3x) is 4ln|x|-3/2 ln|x^2+3|-1/(sqrt(3))arctan(x/(sqrt(3)))+C

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integral of (x^2-x+12)/(x^3+3x)

\int \frac{x ^{2} - x + 12}{x ^{3} + 3 x} d x

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

Solution steps

\int \frac{x ^{2} - x + 12}{x ^{3} + 3 x} d x

Take the partial fraction of

\frac{x ^{2} - x + 12}{x ^{3} + 3 x} : \frac{4}{x} + \frac{- 3 x - 1}{x ^{2} + 3}

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

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

Add a constant to the solution

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

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

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

y = x arcsin (x) + 1 - x^{2}

\frac{\partial}{\partial x} (9 sin (2 x + y))

\frac{\partial}{\partial x} (x^{2} y + y z^{3} + z^{5} x)

What is the integral of (x^2-x+12)/(x^3+3x) ?
The integral of (x^2-x+12)/(x^3+3x) is 4ln|x|-3/2 ln|x^2+3|-1/(sqrt(3))arctan(x/(sqrt(3)))+C