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

The integral of ((x^3-4+x))/(x-2) is 3x^2+x+6ln|x-2|+1/3 (x-2)^3-14+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

3 x^{2} + x + 6 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 14 + C

Solution steps

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

Expand

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

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

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

Simplify

= 3 x^{2} + x + 6 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 14

Add a constant to the solution

= 3 x^{2} + x + 6 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 14 + C

\int 18 tan^{4} (x) d x

\int (x + 7) (x - x + 1) d x

\int cos^{4} (6 x) sin (6 x) d x

\int \frac{4 x ^{3}}{x ^{4} - 3} d x

\int sin^{2} (3 u) d u

What is the integral of ((x^3-4+x))/(x-2) ?
The integral of ((x^3-4+x))/(x-2) is 3x^2+x+6ln|x-2|+1/3 (x-2)^3-14+C