What is the integral of 1/(x(2000-x)) ?

The integral of 1/(x(2000-x)) is 1/2000 ln|x|-1/2000 ln|x-2000|+C

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integral of 1/(x(2000-x))

\int \frac{1}{x ( 2000 - x )} d x

\frac{1}{2000} ln ∣ x ∣ - \frac{1}{2000} ln ∣ x - 2000 ∣ + C

Solution steps

\int \frac{1}{x ( 2000 - x )} d x

Take the partial fraction of

\frac{1}{x ( 2000 - x )} : \frac{1}{2000 x} - \frac{1}{2000 ( x - 2000 )}

= \int \frac{1}{2000 x} - \frac{1}{2000 ( x - 2000 )} 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}{2000 x} d x - \int \frac{1}{2000 ( x - 2000 )} d x

\int \frac{1}{2000 x} d x = \frac{1}{2000} ln ∣ x ∣

\int \frac{1}{2000 ( x - 2000 )} d x = \frac{1}{2000} ln ∣ x - 2000 ∣

= \frac{1}{2000} ln ∣ x ∣ - \frac{1}{2000} ln ∣ x - 2000 ∣

Add a constant to the solution

= \frac{1}{2000} ln ∣ x ∣ - \frac{1}{2000} ln ∣ x - 2000 ∣ + C

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What is the integral of 1/(x(2000-x)) ?
The integral of 1/(x(2000-x)) is 1/2000 ln|x|-1/2000 ln|x-2000|+C