What is the integral of (x^2+20x-15)/(x^3+4x^2-5x) ?

The integral of (x^2+20x-15)/(x^3+4x^2-5x) is 3ln|x|+ln|x-1|-3ln|x+5|+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+20x-15)/(x^3+4x^2-5x)

\int \frac{x ^{2} + 20 x - 15}{x ^{3} + 4 x ^{2} - 5 x} d x

3 ln ∣ x ∣ + ln ∣ x - 1 ∣ - 3 ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{x ^{2} + 20 x - 15}{x ^{3} + 4 x ^{2} - 5 x} d x

Take the partial fraction of

\frac{x ^{2} + 20 x - 15}{x ^{3} + 4 x ^{2} - 5 x} : \frac{3}{x} + \frac{1}{x - 1} - \frac{3}{x + 5}

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

= 3 ln ∣ x ∣ + ln ∣ x - 1 ∣ - 3 ln ∣ x + 5 ∣

Add a constant to the solution

= 3 ln ∣ x ∣ + ln ∣ x - 1 ∣ - 3 ln ∣ x + 5 ∣ + C

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

\frac{d y}{d x} = \frac{9 y ^{2}}{x}

\int_{1}^{R} x^{- \frac{9}{4}} d x

n = 1 \sum \infty \frac{1}{5 ^{n} + 7}

ln (7 - 3 x)

What is the integral of (x^2+20x-15)/(x^3+4x^2-5x) ?
The integral of (x^2+20x-15)/(x^3+4x^2-5x) is 3ln|x|+ln|x-1|-3ln|x+5|+C