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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Add a constant to the solution

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

5 x^{3}

\int x sin^{2} (3 x) d x

\int 3 t d t

1 - x x

\frac{\partial}{\partial z} (y cos (z))

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