What is the integral of (4x)/(x-2) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (4x)/(x-2)

\int \frac{4 x}{x - 2} d x

4 (x - 2 + 2 ln ∣ x - 2 ∣) + C

Solution steps

\int \frac{4 x}{x - 2} d x

Take the constant out:

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

= 4 \cdot \int \frac{x}{x - 2} d x

Apply u-substitution

: \int \frac{u + 2}{u} d u

= 4 \cdot \int \frac{u + 2}{u} d u

Expand

\frac{u + 2}{u} : 1 + \frac{2}{u}

= 4 \cdot \int 1 + \frac{2}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (\int 1 d u + \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= 4 (u + 2 ln ∣ u ∣)

Substitute back

u = x - 2

= 4 (x - 2 + 2 ln ∣ x - 2 ∣)

Add a constant to the solution

= 4 (x - 2 + 2 ln ∣ x - 2 ∣) + C

\int 2 x y^{3} + 1 d x

\int x (1 + x^{2})^{9} d x

\int ln (10)

\int 3 x e^{\frac{1}{2} x^{2}} d x

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