What is the integral of (x^2-x+18)/(x^3+3x) ?

The integral of (x^2-x+18)/(x^3+3x) is 6ln|x|-5/2 ln|x^2+3|-1/(sqrt(3))arctan(x/(sqrt(3)))+C

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integral of (x^2-x+18)/(x^3+3x)

\int \frac{x ^{2} - x + 18}{x ^{3} + 3 x} d x

6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3}) + C

Solution steps

\int \frac{x ^{2} - x + 18}{x ^{3} + 3 x} d x

Take the partial fraction of

\frac{x ^{2} - x + 18}{x ^{3} + 3 x} : \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 3}

= \int \frac{6}{x} + \frac{- 5 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{6}{x} d x + \int \frac{- 5 x - 1}{x ^{2} + 3} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{- 5 x - 1}{x ^{2} + 3} d x = - \frac{5}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3})

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3})

Add a constant to the solution

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3}) + C

\int \frac{1}{10 x - x ^{2}} d x

\frac{d}{d x} (0.2 e^{5 x})

f (x) = sin^{5} (3 x^{2} - x + 13)

\frac{81}{( 6 ( \frac{1}{6} ) + 8 ) ^{\frac{5}{2}}}

(arctan (5 x))^{2}

What is the integral of (x^2-x+18)/(x^3+3x) ?
The integral of (x^2-x+18)/(x^3+3x) is 6ln|x|-5/2 ln|x^2+3|-1/(sqrt(3))arctan(x/(sqrt(3)))+C