What is the integral of (x^2+30x-9)/(x^3-9x) ?

The integral of (x^2+30x-9)/(x^3-9x) is ln|x|-5ln|x+3|+5ln|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 (x^2+30x-9)/(x^3-9x)

\int \frac{x ^{2} + 30 x - 9}{x ^{3} - 9 x} d x

ln ∣ x ∣ - 5 ln ∣ x + 3 ∣ + 5 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x ^{2} + 30 x - 9}{x ^{3} - 9 x} d x

Take the partial fraction of

\frac{x ^{2} + 30 x - 9}{x ^{3} - 9 x} : \frac{1}{x} - \frac{5}{x + 3} + \frac{5}{x - 3}

= \int \frac{1}{x} - \frac{5}{x + 3} + \frac{5}{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{5}{x + 3} d x + \int \frac{5}{x - 3} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{5}{x + 3} d x = 5 ln ∣ x + 3 ∣

\int \frac{5}{x - 3} d x = 5 ln ∣ x - 3 ∣

= ln ∣ x ∣ - 5 ln ∣ x + 3 ∣ + 5 ln ∣ x - 3 ∣

Add a constant to the solution

= ln ∣ x ∣ - 5 ln ∣ x + 3 ∣ + 5 ln ∣ x - 3 ∣ + C

\int e^{3 x} (x + e^{- 2 x}) d x

\int \frac{1}{4 x ^{2} + c ^{2}} d x

\int \frac{7 x ^{2}}{( 81 + x ^{2} ) ^{2}} d x

\int \frac{19}{( x ^{2} - 1 ) ^{2}} d x

\int \frac{arcsin ( x )}{2 x ^{2}} d x

What is the integral of (x^2+30x-9)/(x^3-9x) ?
The integral of (x^2+30x-9)/(x^3-9x) is ln|x|-5ln|x+3|+5ln|x-3|+C