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

The integral of-1/(x(x-1)) is 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/(x(x-1))

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

ln ∣ x ∣ - ln ∣ x - 1 ∣ + C

Solution steps

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

Take the constant out:

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

= - \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}

= - \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

= - (- \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 ∣

= - (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

Simplify

= ln ∣ x ∣ - ln ∣ x - 1 ∣

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ x - 1 ∣ + C

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

\int \frac{x ^{3}}{( 9 ) ^{2} - x ^{2}} d x

\int r^{2} r d r

\int 7 π d x

\int \frac{x}{3 x ^{2} - 5} d x

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