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

The integral of ((x^5+x^2-x+5)/x) is (x^5)/5+(x^2)/2-x+5ln|x|+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^2-x+5)/x)

\int (\frac{x ^{5} + x ^{2} - x + 5}{x}) d x

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

Solution steps

\int \frac{x ^{5} + x ^{2} - x + 5}{x} d x

Expand

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

= \int x^{4} + x - 1 + \frac{5}{x} 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 d x - \int 1 d x + \int \frac{5}{x} d x

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

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

\int 1 d x = x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

Add a constant to the solution

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

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