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

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

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

ln ∣ x - 1 ∣ + \frac{1}{x - 1} + C

Solution steps

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

Expand

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

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

\int \frac{x}{( x - 1 ) ^{2}} d x = ln ∣ x - 1 ∣ - \frac{1}{x - 1}

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

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

Simplify

= ln ∣ x - 1 ∣ + \frac{1}{x - 1}

Add a constant to the solution

= ln ∣ x - 1 ∣ + \frac{1}{x - 1} + C

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

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

\int x^{- 1.5} d x

\int \frac{y - 1}{e ^{y}} d y

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

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