What is the integral of (x^2+3x-5)/(x-3) ?

The integral of (x^2+3x-5)/(x-3) is 9x+13ln|x-3|+1/2 (x-3)^2-27+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^2+3x-5)/(x-3)

\int \frac{x ^{2} + 3 x - 5}{x - 3} d x

9 x + 13 ln ∣ x - 3 ∣ + \frac{1}{2} (x - 3)^{2} - 27 + C

Solution steps

\int \frac{x ^{2} + 3 x - 5}{x - 3} d x

Expand

\frac{x ^{2} + 3 x - 5}{x - 3} : \frac{x ^{2}}{x - 3} + \frac{3 x}{x - 3} - \frac{5}{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 ^{2}}{x - 3} d x + \int \frac{3 x}{x - 3} d x - \int \frac{5}{x - 3} d x

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

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

\int \frac{5}{x - 3} d x = 5 ln ∣ x - 3 ∣

= \frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣ + 3 (x - 3 + 3 ln ∣ x - 3 ∣) - 5 ln ∣ x - 3 ∣

Simplify

\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣ + 3 (x - 3 + 3 ln ∣ x - 3 ∣) - 5 ln ∣ x - 3 ∣ : 9 x + 13 ln ∣ x - 3 ∣ + \frac{1}{2} (x - 3)^{2} - 27

= 9 x + 13 ln ∣ x - 3 ∣ + \frac{1}{2} (x - 3)^{2} - 27

Add a constant to the solution

= 9 x + 13 ln ∣ x - 3 ∣ + \frac{1}{2} (x - 3)^{2} - 27 + C

x \to 5 + lim (\frac{2 - x}{x - 5})

\int - 4 x (- 6 x^{2} - 1)^{6} d x

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

\frac{\partial}{\partial x} (x e^{x})

\frac{d}{d x} (2 x^{2} e^{\frac{x}{2}})

What is the integral of (x^2+3x-5)/(x-3) ?
The integral of (x^2+3x-5)/(x-3) is 9x+13ln|x-3|+1/2 (x-3)^2-27+C