What is the integral of 1/(x(1000-x)) ?

The integral of 1/(x(1000-x)) is 1/1000 ln|x|-1/1000 ln|x-1000|+C

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integral of 1/(x(1000-x))

\int \frac{1}{x ( 1000 - x )} d x

\frac{1}{1000} ln ∣ x ∣ - \frac{1}{1000} ln ∣ x - 1000 ∣ + C

Solution steps

\int \frac{1}{x ( 1000 - x )} d x

Take the partial fraction of

\frac{1}{x ( 1000 - x )} : \frac{1}{1000 x} - \frac{1}{1000 ( x - 1000 )}

= \int \frac{1}{1000 x} - \frac{1}{1000 ( x - 1000 )} 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}{1000 x} d x - \int \frac{1}{1000 ( x - 1000 )} d x

\int \frac{1}{1000 x} d x = \frac{1}{1000} ln ∣ x ∣

\int \frac{1}{1000 ( x - 1000 )} d x = \frac{1}{1000} ln ∣ x - 1000 ∣

= \frac{1}{1000} ln ∣ x ∣ - \frac{1}{1000} ln ∣ x - 1000 ∣

Add a constant to the solution

= \frac{1}{1000} ln ∣ x ∣ - \frac{1}{1000} ln ∣ x - 1000 ∣ + C

{\frac{1}{s ( s - 1 )}}

f (x) = cos (\frac{π x}{2})

\int \frac{x}{1 - x x} d x

y = (x + 3)^{2} - (x - 2)^{2}

(3, - 6), (5, - 5)

What is the integral of 1/(x(1000-x)) ?
The integral of 1/(x(1000-x)) is 1/1000 ln|x|-1/1000 ln|x-1000|+C