What is the integral of (x+1)/(x^3+5x^2+6x) ?

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

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integral of (x+1)/(x^3+5x^2+6x)

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

\frac{1}{6} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ - \frac{2}{3} ln ∣ x + 3 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \int \frac{1}{6 x} + \frac{1}{2 ( x + 2 )} - \frac{2}{3 ( x + 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{1}{6 x} d x + \int \frac{1}{2 ( x + 2 )} d x - \int \frac{2}{3 ( x + 3 )} d x

\int \frac{1}{6 x} d x = \frac{1}{6} ln ∣ x ∣

\int \frac{1}{2 ( x + 2 )} d x = \frac{1}{2} ln ∣ x + 2 ∣

\int \frac{2}{3 ( x + 3 )} d x = \frac{2}{3} ln ∣ x + 3 ∣

= \frac{1}{6} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ - \frac{2}{3} ln ∣ x + 3 ∣

Add a constant to the solution

= \frac{1}{6} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ - \frac{2}{3} ln ∣ x + 3 ∣ + C

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