What is the integral of ((8x^5-3x^2)/(2x^3)) ?

The integral of ((8x^5-3x^2)/(2x^3)) is 1/2 ((8x^3)/3-3ln|x|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of ((8x^5-3x^2)/(2x^3))

\int (\frac{8 x ^{5} - 3 x ^{2}}{2 x ^{3}}) d x

\frac{1}{2} (\frac{8 x ^{3}}{3} - 3 ln ∣ x ∣) + C

Solution steps

\int \frac{8 x ^{5} - 3 x ^{2}}{2 x ^{3}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{8 x ^{5} - 3 x ^{2}}{x ^{3}} d x

Simplify

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

= \frac{1}{2} \cdot \int \frac{8 x ^{3} - 3}{x} d x

Expand

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

= \frac{1}{2} \cdot \int 8 x^{2} - \frac{3}{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

= \frac{1}{2} (\int 8 x^{2} d x - \int \frac{3}{x} d x)

\int 8 x^{2} d x = \frac{8 x ^{3}}{3}

\int \frac{3}{x} d x = 3 ln ∣ x ∣

= \frac{1}{2} (\frac{8 x ^{3}}{3} - 3 ln ∣ x ∣)

Add a constant to the solution

= \frac{1}{2} (\frac{8 x ^{3}}{3} - 3 ln ∣ x ∣) + C

What is the integral of ((8x^5-3x^2)/(2x^3)) ?
The integral of ((8x^5-3x^2)/(2x^3)) is 1/2 ((8x^3)/3-3ln|x|)+C