What is the integral of (2x)/((x-1)(x-2)) ?

The integral of (2x)/((x-1)(x-2)) is 2(-ln|x-1|+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 (2x)/((x-1)(x-2))

\int \frac{2 x}{( x - 1 ) ( x - 2 )} d x

2 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣) + C

Solution steps

\int \frac{2 x}{( x - 1 ) ( x - 2 )} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{( x - 1 ) ( x - 2 )} d x

Take the partial fraction of

\frac{x}{( x - 1 ) ( x - 2 )} : - \frac{1}{x - 1} + \frac{2}{x - 2}

= 2 \cdot \int - \frac{1}{x - 1} + \frac{2}{x - 2} 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

= 2 (- \int \frac{1}{x - 1} d x + \int \frac{2}{x - 2} d x)

\int \frac{1}{x - 1} d x = ln ∣ x - 1 ∣

\int \frac{2}{x - 2} d x = 2 ln ∣ x - 2 ∣

= 2 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣)

Add a constant to the solution

= 2 (- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣) + C

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

\frac{d}{d x} (\frac{4 y ^{2} - x ^{4}}{4 x y})

\int \frac{2 x}{x ^{2} - 49} d x

ln (3 - x), 2

\int \frac{sec ^{2} ( x )}{x} d x

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