What is the integral of (x^5+x)/(x+1) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x^5+x)/(x+1)

\int \frac{x ^{5} + x}{x + 1} d x

\frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - 2 ln ∣ x + 1 ∣ + 1 + C

Solution steps

\int \frac{x ^{5} + x}{x + 1} d x

Expand

\frac{x ^{5} + x}{x + 1} : \frac{x ^{5}}{x + 1} + \frac{x}{x + 1}

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 ^{5}}{x + 1} d x + \int \frac{x}{x + 1} d x

\int \frac{x ^{5}}{x + 1} d x = \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x - ln ∣ x + 1 ∣

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

= \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x - ln ∣ x + 1 ∣ + x + 1 - ln ∣ x + 1 ∣

Simplify

= \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - 2 ln ∣ x + 1 ∣ + 1

Add a constant to the solution

= \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - 2 ln ∣ x + 1 ∣ + 1 + C

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