What is the integral of (2x^2-9x-9)/(x^3-9x) ?

The integral of (2x^2-9x-9)/(x^3-9x) is ln|x|+2ln|x+3|-ln|x-3|+C

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integral of (2x^2-9x-9)/(x^3-9x)

\int \frac{2 x ^{2} - 9 x - 9}{x ^{3} - 9 x} d x

ln ∣ x ∣ + 2 ln ∣ x + 3 ∣ - ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{2 x ^{2} - 9 x - 9}{x ^{3} - 9 x} d x

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int \frac{1}{x - 3} d x = ln ∣ x - 3 ∣

= ln ∣ x ∣ + 2 ln ∣ x + 3 ∣ - ln ∣ x - 3 ∣

Add a constant to the solution

= ln ∣ x ∣ + 2 ln ∣ x + 3 ∣ - ln ∣ x - 3 ∣ + C

\int (I n x)^{2} d x

\int cos (x) + (x^{2} + 1)^{7} \cdot 2 x d x

\frac{d y}{d x}, ln (x y^{3}) = x y

\int \frac{tan ^{3} ( ln ( x ) )}{2 x} d x

f (x) = cos^{5} (x)

What is the integral of (2x^2-9x-9)/(x^3-9x) ?
The integral of (2x^2-9x-9)/(x^3-9x) is ln|x|+2ln|x+3|-ln|x-3|+C