What is the integral of (2x-1)/((x-3)^2) ?

The integral of (2x-1)/((x-3)^2) is 2ln|x-3|-5/(x-3)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2x-1)/((x-3)^2)

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

2 ln ∣ x - 3 ∣ - \frac{5}{x - 3} + C

Solution steps

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

Expand

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

\int \frac{2 x}{( x - 3 ) ^{2}} d x = 2 (ln ∣ x - 3 ∣ - \frac{3}{x - 3})

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

= 2 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - (- \frac{1}{x - 3})

Simplify

2 (ln ∣ x - 3 ∣ - \frac{3}{x - 3}) - (- \frac{1}{x - 3}) : 2 ln ∣ x - 3 ∣ - \frac{5}{x - 3}

= 2 ln ∣ x - 3 ∣ - \frac{5}{x - 3}

Add a constant to the solution

= 2 ln ∣ x - 3 ∣ - \frac{5}{x - 3} + C

\int \frac{- 7 x + 1}{( x + 1 ) ( x - 1 ) ^{2}} d x

\int \frac{9 cosh ( x )}{x} d x

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

\int \frac{1}{P ( 1 - \frac{P}{K} )} d P

\int x y cos (y) d y

What is the integral of (2x-1)/((x-3)^2) ?
The integral of (2x-1)/((x-3)^2) is 2ln|x-3|-5/(x-3)+C