What is the integral of x/((x-2)(x^2-4x+5)^2) ?

The integral of x/((x-2)(x^2-4x+5)^2) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x/((x-2)(x^2-4x+5)^2)

\int \frac{x}{( x - 2 ) ( x ^{2} - 4 x + 5 ) ^{2}} d x

2 ln ∣ x - 2 ∣ - ln x^{2} - 4 x + 5 + \frac{1}{2} arctan (x - 2) + \frac{5}{4} sin (2 arctan (x - 2)) + \frac{1}{x ^{2} - 4 x + 5} - \frac{2 ( x - 2 )}{x ^{2} - 4 x + 5} + C

Solution steps

\int \frac{x}{( x - 2 ) ( x ^{2} - 4 x + 5 ) ^{2}} d x

Take the partial fraction of

\frac{x}{( x - 2 ) ( x ^{2} - 4 x + 5 ) ^{2}} : \frac{2}{x - 2} + \frac{- 2 x + 4}{x ^{2} - 4 x + 5} + \frac{- 2 x + 5}{( x ^{2} - 4 x + 5 ) ^{2}}

= \int \frac{2}{x - 2} + \frac{- 2 x + 4}{x ^{2} - 4 x + 5} + \frac{- 2 x + 5}{( x ^{2} - 4 x + 5 ) ^{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{2}{x - 2} d x + \int \frac{- 2 x + 4}{x ^{2} - 4 x + 5} d x + \int \frac{- 2 x + 5}{( x ^{2} - 4 x + 5 ) ^{2}} d x

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

\int \frac{- 2 x + 4}{x ^{2} - 4 x + 5} d x = - ln x^{2} - 4 x + 5

\int \frac{- 2 x + 5}{( x ^{2} - 4 x + 5 ) ^{2}} d x = \frac{1}{2} arctan (x - 2) + \frac{5}{4} sin (2 arctan (x - 2)) + \frac{1}{x ^{2} - 4 x + 5} - \frac{2 ( x - 2 )}{x ^{2} - 4 x + 5}

= 2 ln ∣ x - 2 ∣ - ln x^{2} - 4 x + 5 + \frac{1}{2} arctan (x - 2) + \frac{5}{4} sin (2 arctan (x - 2)) + \frac{1}{x ^{2} - 4 x + 5} - \frac{2 ( x - 2 )}{x ^{2} - 4 x + 5}

Add a constant to the solution

= 2 ln ∣ x - 2 ∣ - ln x^{2} - 4 x + 5 + \frac{1}{2} arctan (x - 2) + \frac{5}{4} sin (2 arctan (x - 2)) + \frac{1}{x ^{2} - 4 x + 5} - \frac{2 ( x - 2 )}{x ^{2} - 4 x + 5} + C

x \to 2 lim (x^{5} - 8 - 2)

\frac{d}{d x} (x^{- 3} + x^{- 5} - 2 x^{- 6})

\int 5 cot^{3} (x) d x

\int_{1}^{3} 3 x^{4} ln (x) d x

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

What is the integral of x/((x-2)(x^2-4x+5)^2) ?
The integral of x/((x-2)(x^2-4x+5)^2) is