What is the integral of (x-4)/(x(x-2)(x-5)) ?

The integral of (x-4)/(x(x-2)(x-5)) is 1/3 ln|x-2|+1/15 ln|x-5|-2/5 ln|x|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x-4)/(x(x-2)(x-5))

\int \frac{x - 4}{x ( x - 2 ) ( x - 5 )} d x

\frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣ - \frac{2}{5} ln ∣ x ∣ + C

Solution steps

\int \frac{x - 4}{x ( x - 2 ) ( x - 5 )} d x

Expand

\frac{x - 4}{x ( x - 2 ) ( x - 5 )} : \frac{x}{x ( x - 2 ) ( x - 5 )} - \frac{4}{x ( x - 2 ) ( x - 5 )}

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}{x ( x - 2 ) ( x - 5 )} d x - \int \frac{4}{x ( x - 2 ) ( x - 5 )} d x

\int \frac{x}{x ( x - 2 ) ( x - 5 )} d x = - \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x - 5 ∣

\int \frac{4}{x ( x - 2 ) ( x - 5 )} d x = 4 (\frac{1}{10} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣)

= - \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x - 5 ∣ - 4 (\frac{1}{10} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣)

Simplify

- \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{3} ln ∣ x - 5 ∣ - 4 (\frac{1}{10} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣) : \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣ - \frac{2}{5} ln ∣ x ∣

= \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣ - \frac{2}{5} ln ∣ x ∣

Add a constant to the solution

= \frac{1}{3} ln ∣ x - 2 ∣ + \frac{1}{15} ln ∣ x - 5 ∣ - \frac{2}{5} ln ∣ x ∣ + C

\int \frac{40}{S ^{2}} d S

\int \frac{2}{( 5 - 3 x ) ^{9}} d x

\int ln (2 x - 3) d x

\int (x^{3} - x^{2} - x + 1) e^{2 x} d x

\int \frac{1}{4 x ^{2} + 8 x + 19} d x

What is the integral of (x-4)/(x(x-2)(x-5)) ?
The integral of (x-4)/(x(x-2)(x-5)) is 1/3 ln|x-2|+1/15 ln|x-5|-2/5 ln|x|+C