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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

\int \frac{( x ^{5} - 1 )}{( x - 1 )} d x

\frac{x ^{5}}{5} + \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x + C

Solution steps

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

Long division

\frac{x ^{5} - 1}{x - 1} : x^{4} + x^{3} + x^{2} + x + 1

= \int x^{4} + x^{3} + x^{2} + 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^{4} d x + \int x^{3} d x + \int x^{2} d x + \int x d x + \int 1 d x

\int x^{4} d x = \frac{x ^{5}}{5}

\int x^{3} d x = \frac{x ^{4}}{4}

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

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

\int 1 d x = x

= \frac{x ^{5}}{5} + \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x

Add a constant to the solution

= \frac{x ^{5}}{5} + \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + x + C

\int \frac{2 x - 3}{x ^{2} - 8 x + 7} d x

\int t t - 6 d t

\int 9 x (6 - x) d x

\int \frac{x}{( x ^{2} + 9 )} d x

\int 2 u d u

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