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

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

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

- ln ∣ x - 1 ∣ + ln x^{2} + 2 x + 2 + arctan (x + 1) + C

Solution steps

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

Take the partial fraction of

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

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

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

\int \frac{2 x + 3}{x ^{2} + 2 x + 2} d x = ln x^{2} + 2 x + 2 + arctan (x + 1)

= - ln ∣ x - 1 ∣ + ln x^{2} + 2 x + 2 + arctan (x + 1)

Add a constant to the solution

= - ln ∣ x - 1 ∣ + ln x^{2} + 2 x + 2 + arctan (x + 1) + C

\frac{7}{sin ( x ) + cos ( x )}

\int - 9 e^{- 9 x} + \frac{- 7 x + 5 x ^{5}}{x ^{2}} d x

x \to 2 lim (3 x^{2} + 7 x - 15)

\int (\frac{a}{b x} + \frac{x}{c ^{2} x + x ^{3}}) d x

\frac{d y}{d x} + 3 y = 0

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