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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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 e^{3 x} (x^{2} + y^{2}) d y

\int 6 y^{2} - x^{2} + 3 d y

\int \frac{1}{x} - \frac{2}{x ^{3}} d x

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

\int \frac{1}{5 x + 2} d 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