What is the integral of (80)/(x(20-x)) ?

The integral of (80)/(x(20-x)) is 80(1/20 ln|x|-1/20 ln|x-20|)+C

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integral of (80)/(x(20-x))

\int \frac{80}{x ( 20 - x )} d x

80 (\frac{1}{20} ln ∣ x ∣ - \frac{1}{20} ln ∣ x - 20 ∣) + C

Solution steps

\int \frac{80}{x ( 20 - x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 80 \cdot \int \frac{1}{x ( 20 - x )} d x

Take the partial fraction of

\frac{1}{x ( 20 - x )} : \frac{1}{20 x} - \frac{1}{20 ( x - 20 )}

= 80 \cdot \int \frac{1}{20 x} - \frac{1}{20 ( x - 20 )} 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

= 80 (\int \frac{1}{20 x} d x - \int \frac{1}{20 ( x - 20 )} d x)

\int \frac{1}{20 x} d x = \frac{1}{20} ln ∣ x ∣

\int \frac{1}{20 ( x - 20 )} d x = \frac{1}{20} ln ∣ x - 20 ∣

= 80 (\frac{1}{20} ln ∣ x ∣ - \frac{1}{20} ln ∣ x - 20 ∣)

Add a constant to the solution

= 80 (\frac{1}{20} ln ∣ x ∣ - \frac{1}{20} ln ∣ x - 20 ∣) + C

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What is the integral of (80)/(x(20-x)) ?
The integral of (80)/(x(20-x)) is 80(1/20 ln|x|-1/20 ln|x-20|)+C