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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (1/2 x)/(2-x)

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

- ln ∣ 2 - x ∣ - \frac{1}{2} x + C

Solution steps

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

\frac{x}{2 - x} = \frac{2}{2 - x} - 1

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

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

\int 1 d x = x

= \frac{1}{2} (- 2 ln ∣ 2 - x ∣ - x)

Simplify

\frac{1}{2} (- 2 ln ∣ 2 - x ∣ - x) : - ln ∣ 2 - x ∣ - \frac{1}{2} x

= - ln ∣ 2 - x ∣ - \frac{1}{2} x

Add a constant to the solution

= - ln ∣ 2 - x ∣ - \frac{1}{2} x + C

\int \frac{1}{3 x} + \frac{1}{1 - 2 x} d x

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

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

\int (cos (x) - sec^{2} (x)) d x

\int f (y) d y

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