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

The integral of (x^2-1)/((x-3)(x-2)^2) is 8ln|x-3|-7ln|x-2|+3/(x-2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

8 ln ∣ x - 3 ∣ - 7 ln ∣ x - 2 ∣ + \frac{3}{x - 2} + C

Solution steps

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

Take the partial fraction of

\frac{x ^{2} - 1}{( x - 3 ) ( x - 2 ) ^{2}} : \frac{8}{x - 3} - \frac{7}{x - 2} - \frac{3}{( x - 2 ) ^{2}}

= \int \frac{8}{x - 3} - \frac{7}{x - 2} - \frac{3}{( x - 2 ) ^{2}} 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{8}{x - 3} d x - \int \frac{7}{x - 2} d x - \int \frac{3}{( x - 2 ) ^{2}} d x

\int \frac{8}{x - 3} d x = 8 ln ∣ x - 3 ∣

\int \frac{7}{x - 2} d x = 7 ln ∣ x - 2 ∣

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

= 8 ln ∣ x - 3 ∣ - 7 ln ∣ x - 2 ∣ - (- \frac{3}{x - 2})

Simplify

= 8 ln ∣ x - 3 ∣ - 7 ln ∣ x - 2 ∣ + \frac{3}{x - 2}

Add a constant to the solution

= 8 ln ∣ x - 3 ∣ - 7 ln ∣ x - 2 ∣ + \frac{3}{x - 2} + C

\frac{d}{d x} (\frac{4 x}{x ^{2} + 2})

\frac{\partial}{\partial x} (x y - z^{2})

\int (8 - \frac{( 5 ^{x} )}{2}) d x

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

\int t^{2} t^{3} - 1 d t

What is the integral of (x^2-1)/((x-3)(x-2)^2) ?
The integral of (x^2-1)/((x-3)(x-2)^2) is 8ln|x-3|-7ln|x-2|+3/(x-2)+C