What is the integral of 1/(x(5-x)) ?

The integral of 1/(x(5-x)) is 1/5 ln|x|-1/5 ln|x-5|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(x(5-x))

\int \frac{1}{x ( 5 - x )} d x

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

Solution steps

\int \frac{1}{x ( 5 - x )} d x

Take the partial fraction of

\frac{1}{x ( 5 - x )} : \frac{1}{5 x} - \frac{1}{5 ( x - 5 )}

= \int \frac{1}{5 x} - \frac{1}{5 ( x - 5 )} d x

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{1}{5 x} d x - \int \frac{1}{5 ( x - 5 )} d x

\int \frac{1}{5 x} d x = \frac{1}{5} ln ∣ x ∣

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

= \frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x - 5 ∣

Add a constant to the solution

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

\int \frac{ln ( x + 9 )}{x + 9} d x

\int_{- 1}^{9} (9 - x^{2}) - (- 8 x) d x

y^{^{'}} = - 3 y^{2}

\frac{d}{d x} (\frac{1}{12} (x + 3)^{3} + (x + 3)^{- 1})

\int_{1}^{4} t ln (\frac{t}{2}) d t

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