What is the integral of (2x^4+3x)/(2x^2) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2x^4+3x)/(2x^2)

\int \frac{2 x ^{4} + 3 x}{2 x ^{2}} d x

\frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣ + C

Solution steps

\int \frac{2 x ^{4} + 3 x}{2 x ^{2}} 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{2 x ^{4} + 3 x}{x ^{2}} d x

Simplify

\frac{2 x ^{4} + 3 x}{x ^{2}} : \frac{2 x ^{3} + 3}{x}

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

Expand

\frac{2 x ^{3} + 3}{x} : 2 x^{2} + \frac{3}{x}

= \frac{1}{2} \cdot \int 2 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 2 x^{2} d x + \int \frac{3}{x} d x)

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

Simplify

\frac{1}{2} (\frac{2 x ^{3}}{3} + 3 ln ∣ x ∣) : \frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣

= \frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{3}{2} ln ∣ x ∣ + C

\int \frac{1}{1 + t} d t

\frac{4 x}{x ^{2} + 4}

\int e^{x^{2}} 4 x d x

\int_{1}^{3} x^{2} d x

x \to 4 + lim (\frac{- 3}{8 - 2 x})

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