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

The integral of (x^3-3x^2+2x-1)/(x-2) is 3x^2-10x+1/3 (x-2)^3-ln|x-2|-3/2 (x-2)^2+8+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

3 x^{2} - 10 x + \frac{1}{3} (x - 2)^{3} - ln ∣ x - 2 ∣ - \frac{3}{2} (x - 2)^{2} + 8 + C

Solution steps

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

Expand

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

\int \frac{x ^{3}}{x - 2} d x = 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12

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

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

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

= 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 - 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + 2 (x - 2 + 2 ln ∣ x - 2 ∣) - ln ∣ x - 2 ∣

Simplify

3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 - 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + 2 (x - 2 + 2 ln ∣ x - 2 ∣) - ln ∣ x - 2 ∣ : 3 x^{2} - 10 x + \frac{1}{3} (x - 2)^{3} - ln ∣ x - 2 ∣ - \frac{3}{2} (x - 2)^{2} + 8

= 3 x^{2} - 10 x + \frac{1}{3} (x - 2)^{3} - ln ∣ x - 2 ∣ - \frac{3}{2} (x - 2)^{2} + 8

Add a constant to the solution

= 3 x^{2} - 10 x + \frac{1}{3} (x - 2)^{3} - ln ∣ x - 2 ∣ - \frac{3}{2} (x - 2)^{2} + 8 + C

What is the integral of (x^3-3x^2+2x-1)/(x-2) ?
The integral of (x^3-3x^2+2x-1)/(x-2) is 3x^2-10x+1/3 (x-2)^3-ln|x-2|-3/2 (x-2)^2+8+C