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

The integral of 1/(N(5-N)) is 1/5 ln|N|-1/5 ln|N-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/(N(5-N))

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

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

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

Add a constant to the solution

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

\int cos (x) (2 + sin (x))^{2} d x

\int \frac{3 x}{3 x + 1} d x

\int \frac{2 x + 5}{5 x + 9} d x

\int \frac{( ln ( x ) + 1 )}{x} d x

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