What is the integral of (7x^2+x+12)/(3x^3+12x) ?

The integral of (7x^2+x+12)/(3x^3+12x) is ln|x|+1/3 (2ln|x^2+4|+1/2 arctan(x/2))+C

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integral of (7x^2+x+12)/(3x^3+12x)

\int \frac{7 x ^{2} + x + 12}{3 x ^{3} + 12 x} d x

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

Solution steps

\int \frac{7 x ^{2} + x + 12}{3 x ^{3} + 12 x} d x

Take the partial fraction of

\frac{7 x ^{2} + x + 12}{3 x ^{3} + 12 x} : \frac{1}{x} + \frac{4 x + 1}{3 ( x ^{2} + 4 )}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Add a constant to the solution

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

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What is the integral of (7x^2+x+12)/(3x^3+12x) ?
The integral of (7x^2+x+12)/(3x^3+12x) is ln|x|+1/3 (2ln|x^2+4|+1/2 arctan(x/2))+C