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

The integral of (2x^3+x-1)/(x-2) is 6x^2+x+17ln|x-2|+2/3 (x-2)^3-26+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

6 x^{2} + x + 17 ln ∣ x - 2 ∣ + \frac{2}{3} (x - 2)^{3} - 26 + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

2 (3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12) + x - 2 + 2 ln ∣ x - 2 ∣ - ln ∣ x - 2 ∣ : 6 x^{2} + x + 17 ln ∣ x - 2 ∣ + \frac{2}{3} (x - 2)^{3} - 26

= 6 x^{2} + x + 17 ln ∣ x - 2 ∣ + \frac{2}{3} (x - 2)^{3} - 26

Add a constant to the solution

= 6 x^{2} + x + 17 ln ∣ x - 2 ∣ + \frac{2}{3} (x - 2)^{3} - 26 + C

\int cos (x) sin (x) x d x

\int (lo g_{10} (x))^{2} d x

\int \frac{ln ( x )}{x} e^{l n^{2} (x)} d x

\int (x - 1)^{6} d x

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

What is the integral of (2x^3+x-1)/(x-2) ?
The integral of (2x^3+x-1)/(x-2) is 6x^2+x+17ln|x-2|+2/3 (x-2)^3-26+C