What is the integral of (4x^2+x+27)/(x^3+9x) ?

The integral of (4x^2+x+27)/(x^3+9x) is 3ln|x|+1/2 ln|x^2+9|+1/3 arctan(x/3)+C

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integral of (4x^2+x+27)/(x^3+9x)

\int \frac{4 x ^{2} + x + 27}{x ^{3} + 9 x} d x

3 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3}) + C

Solution steps

\int \frac{4 x ^{2} + x + 27}{x ^{3} + 9 x} d x

Take the partial fraction of

\frac{4 x ^{2} + x + 27}{x ^{3} + 9 x} : \frac{3}{x} + \frac{x + 1}{x ^{2} + 9}

= \int \frac{3}{x} + \frac{x + 1}{x ^{2} + 9} 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{3}{x} d x + \int \frac{x + 1}{x ^{2} + 9} d x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

\int \frac{x + 1}{x ^{2} + 9} d x = \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})

= 3 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})

Add a constant to the solution

= 3 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3}) + C

What is the integral of (4x^2+x+27)/(x^3+9x) ?
The integral of (4x^2+x+27)/(x^3+9x) is 3ln|x|+1/2 ln|x^2+9|+1/3 arctan(x/3)+C