What is the integral of (x^3+x^2)/(x^2+5x+6) ?

The integral of (x^3+x^2)/(x^2+5x+6) is (x^2)/2-4x-4ln|x+2|+18ln|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^3+x^2)/(x^2+5x+6)

\int \frac{x ^{3} + x ^{2}}{x ^{2} + 5 x + 6} d x

\frac{x ^{2}}{2} - 4 x - 4 ln ∣ x + 2 ∣ + 18 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{x ^{3} + x ^{2}}{x ^{2} + 5 x + 6} d x

Long division

\frac{x ^{3} + x ^{2}}{x ^{2} + 5 x + 6} : x - 4 + \frac{14 x + 24}{x ^{2} + 5 x + 6}

= \int x - 4 + \frac{14 x + 24}{x ^{2} + 5 x + 6} 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 x d x - \int 4 d x + \int \frac{14 x + 24}{x ^{2} + 5 x + 6} d x

\int x d x = \frac{x ^{2}}{2}

\int 4 d x = 4 x

\int \frac{14 x + 24}{x ^{2} + 5 x + 6} d x = - 4 ln ∣ x + 2 ∣ + 18 ln ∣ x + 3 ∣

= \frac{x ^{2}}{2} - 4 x - 4 ln ∣ x + 2 ∣ + 18 ln ∣ x + 3 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - 4 x - 4 ln ∣ x + 2 ∣ + 18 ln ∣ x + 3 ∣ + C

\frac{d}{d x} (\frac{x ^{2} e ^{x}}{x ^{2} + e ^{x}})

y^{^{''}} - 4 y^{^{'}} + 4 y^{^{'}} = 0

\int cos^{3} (\frac{x}{12}) d x

x \to 4 + lim (\frac{x - 4}{x - 4})

2 x^{2} - 9 x + 10

What is the integral of (x^3+x^2)/(x^2+5x+6) ?
The integral of (x^3+x^2)/(x^2+5x+6) is (x^2)/2-4x-4ln|x+2|+18ln|x+3|+C