What is the integral of (x^3-2x^2-2x+1)/(x^2-4x+3) ?

The integral of (x^3-2x^2-2x+1)/(x^2-4x+3) is (x^2)/2+2x+6ln|x-1|+7ln|x-3|-5ln|x^2-4x+3|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^3-2x^2-2x+1)/(x^2-4x+3)

\int \frac{x ^{3} - 2 x ^{2} - 2 x + 1}{x ^{2} - 4 x + 3} d x

\frac{x ^{2}}{2} + 2 x + 6 ln ∣ x - 1 ∣ + 7 ln ∣ x - 3 ∣ - 5 ln x^{2} - 4 x + 3 + C

Solution steps

\int \frac{x ^{3} - 2 x ^{2} - 2 x + 1}{x ^{2} - 4 x + 3} d x

Expand

\frac{x ^{3} - 2 x ^{2} - 2 x + 1}{x ^{2} - 4 x + 3} : \frac{x ^{3}}{x ^{2} - 4 x + 3} - \frac{2 x ^{2}}{x ^{2} - 4 x + 3} - \frac{2 x}{x ^{2} - 4 x + 3} + \frac{1}{x ^{2} - 4 x + 3}

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 ^{3}}{x ^{2} - 4 x + 3} d x - \int \frac{2 x ^{2}}{x ^{2} - 4 x + 3} d x - \int \frac{2 x}{x ^{2} - 4 x + 3} d x + \int \frac{1}{x ^{2} - 4 x + 3} d x

\int \frac{x ^{3}}{x ^{2} - 4 x + 3} d x = \frac{x ^{2}}{2} + 4 x - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{27}{2} ln ∣ x - 3 ∣

\int \frac{2 x ^{2}}{x ^{2} - 4 x + 3} d x = 2 (2 ln x^{2} - 4 x + 3 - 5 (\frac{1}{2} ln ∣ x - 1 ∣ - \frac{1}{2} ln ∣ x - 3 ∣) + x)

\int \frac{2 x}{x ^{2} - 4 x + 3} d x = ln x^{2} - 4 x + 3 - 2 ln ∣ x - 1 ∣ + 2 ln ∣ x - 3 ∣

\int \frac{1}{x ^{2} - 4 x + 3} d x = - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} ln ∣ x - 3 ∣

= \frac{x ^{2}}{2} + 4 x - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{27}{2} ln ∣ x - 3 ∣ - 2 (2 ln x^{2} - 4 x + 3 - 5 (\frac{1}{2} ln ∣ x - 1 ∣ - \frac{1}{2} ln ∣ x - 3 ∣) + x) - (ln x^{2} - 4 x + 3 - 2 ln ∣ x - 1 ∣ + 2 ln ∣ x - 3 ∣) - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} ln ∣ x - 3 ∣

Simplify

\frac{x ^{2}}{2} + 4 x - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{27}{2} ln ∣ x - 3 ∣ - 2 (2 ln x^{2} - 4 x + 3 - 5 (\frac{1}{2} ln ∣ x - 1 ∣ - \frac{1}{2} ln ∣ x - 3 ∣) + x) - (ln x^{2} - 4 x + 3 - 2 ln ∣ x - 1 ∣ + 2 ln ∣ x - 3 ∣) - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{1}{2} ln ∣ x - 3 ∣ : \frac{x ^{2}}{2} + 2 x + 6 ln ∣ x - 1 ∣ + 7 ln ∣ x - 3 ∣ - 5 ln x^{2} - 4 x + 3

= \frac{x ^{2}}{2} + 2 x + 6 ln ∣ x - 1 ∣ + 7 ln ∣ x - 3 ∣ - 5 ln x^{2} - 4 x + 3

Add a constant to the solution

= \frac{x ^{2}}{2} + 2 x + 6 ln ∣ x - 1 ∣ + 7 ln ∣ x - 3 ∣ - 5 ln x^{2} - 4 x + 3 + C

\int \frac{a - b}{b x} d x

\int x (1 - x)^{15} d x

\int \frac{5 x ^{2}}{( 81 + x ^{2} ) ^{2}} d x

\int 4 x arcsin (x^{2}) d x

\int \frac{2 x ^{2} - 2 x + 12}{x ^{3} + 3 x} d x

What is the integral of (x^3-2x^2-2x+1)/(x^2-4x+3) ?
The integral of (x^3-2x^2-2x+1)/(x^2-4x+3) is (x^2)/2+2x+6ln|x-1|+7ln|x-3|-5ln|x^2-4x+3|+C