What is the integral of (x^3+4)/(x+2) ?

The integral of (x^3+4)/(x+2) is -3x^2+1/3 (x+2)^3-4ln|x+2|+12+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x^3+4)/(x+2)

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

- 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 4 ln ∣ x + 2 ∣ + 12 + C

Solution steps

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

Expand

\frac{x ^{3} + 4}{x + 2} : \frac{x ^{3}}{x + 2} + \frac{4}{x + 2}

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

\int \frac{x ^{3}}{x + 2} d x = - 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12

\int \frac{4}{x + 2} d x = 4 ln ∣ x + 2 ∣

= - 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12 + 4 ln ∣ x + 2 ∣

Simplify

= - 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 4 ln ∣ x + 2 ∣ + 12

Add a constant to the solution

= - 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 4 ln ∣ x + 2 ∣ + 12 + C

What is the integral of (x^3+4)/(x+2) ?
The integral of (x^3+4)/(x+2) is -3x^2+1/3 (x+2)^3-4ln|x+2|+12+C