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

The integral of (x^3-2x^2+1)/(x-3) is (x^3)/3+(3x^2)/2-x^2+3x+10ln|x-3|+27+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - x^{2} + 3 x + 10 ln ∣ x - 3 ∣ + 27 + C

Solution steps

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

Expand

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

\int \frac{x ^{3}}{x - 3} d x = \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 9 x + 27 ln ∣ x - 3 ∣

\int \frac{2 x ^{2}}{x - 3} d x = x^{2} + 6 x + 18 ln ∣ x - 3 ∣ - 27

\int \frac{1}{x - 3} d x = ln ∣ x - 3 ∣

= \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 9 x + 27 ln ∣ x - 3 ∣ - (x^{2} + 6 x + 18 ln ∣ x - 3 ∣ - 27) + ln ∣ x - 3 ∣

Simplify

\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 9 x + 27 ln ∣ x - 3 ∣ - (x^{2} + 6 x + 18 ln ∣ x - 3 ∣ - 27) + ln ∣ x - 3 ∣ : \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - x^{2} + 3 x + 10 ln ∣ x - 3 ∣ + 27

= \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - x^{2} + 3 x + 10 ln ∣ x - 3 ∣ + 27

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} - x^{2} + 3 x + 10 ln ∣ x - 3 ∣ + 27 + C

\int_{1}^{2} \frac{( ln ( x ) ) ^{2}}{x} d x

\int_{0}^{\infty} u (t) d t

x \to \infty lim (x \cdot 1)

\int t e^{- s t} d t

a re a \frac{17}{4} - \frac{1}{x}, x, \frac{1}{4}, 4

What is the integral of (x^3-2x^2+1)/(x-3) ?
The integral of (x^3-2x^2+1)/(x-3) is (x^3)/3+(3x^2)/2-x^2+3x+10ln|x-3|+27+C