What is the integral of (x^3+2)/(x+1) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x^3+2)/(x+1)

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

\frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x + ln ∣ x + 1 ∣ + C

Solution steps

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

Long division

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

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

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

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

\int 1 d x = x

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

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x + ln ∣ x + 1 ∣

Add a constant to the solution

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x + ln ∣ x + 1 ∣ + C

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