What is the integral of (x^4-8x^2+5x)/(x^2) ?

The integral of (x^4-8x^2+5x)/(x^2) is (x^3)/3-8x+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^4-8x^2+5x)/(x^2)

\int \frac{x ^{4} - 8 x ^{2} + 5 x}{x ^{2}} d x

\frac{x ^{3}}{3} - 8 x + 5 ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{4} - 8 x ^{2} + 5 x}{x ^{2}} d x

Simplify

\frac{x ^{4} - 8 x ^{2} + 5 x}{x ^{2}} : \frac{x ^{3} - 8 x + 5}{x}

= \int \frac{x ^{3} - 8 x + 5}{x} d x

Expand

\frac{x ^{3} - 8 x + 5}{x} : x^{2} - 8 + \frac{5}{x}

= \int x^{2} - 8 + \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^{2} d x - \int 8 d x + \int \frac{5}{x} d x

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

\int 8 d x = 8 x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{x ^{3}}{3} - 8 x + 5 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{3}}{3} - 8 x + 5 ln ∣ x ∣ + C

What is the integral of (x^4-8x^2+5x)/(x^2) ?
The integral of (x^4-8x^2+5x)/(x^2) is (x^3)/3-8x+5ln|x|+C