What is the integral of 1/(x(10-x)) ?

The integral of 1/(x(10-x)) is 1/10 ln|x|-1/10 ln|x-10|+C

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integral of 1/(x(10-x))

\int \frac{1}{x ( 10 - x )} d x

\frac{1}{10} ln ∣ x ∣ - \frac{1}{10} ln ∣ x - 10 ∣ + C

Solution steps

\int \frac{1}{x ( 10 - x )} d x

Take the partial fraction of

\frac{1}{x ( 10 - x )} : \frac{1}{10 x} - \frac{1}{10 ( x - 10 )}

= \int \frac{1}{10 x} - \frac{1}{10 ( x - 10 )} 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}{10 x} d x - \int \frac{1}{10 ( x - 10 )} d x

\int \frac{1}{10 x} d x = \frac{1}{10} ln ∣ x ∣

\int \frac{1}{10 ( x - 10 )} d x = \frac{1}{10} ln ∣ x - 10 ∣

= \frac{1}{10} ln ∣ x ∣ - \frac{1}{10} ln ∣ x - 10 ∣

Add a constant to the solution

= \frac{1}{10} ln ∣ x ∣ - \frac{1}{10} ln ∣ x - 10 ∣ + C

\frac{\partial}{\partial x} (\frac{x + 2}{y + 1})

\frac{d}{d x} (x e^{9 x})

\frac{d}{d x} (\frac{3}{4} x^{\frac{4}{3}} - \frac{3}{8} x^{\frac{2}{3}} + 9)

\int \frac{( a ^{x} - b ^{x} ) ^{2}}{a ^{x} b ^{x}} d x

\frac{d y}{d x} = e^{7 x - 7 y}

What is the integral of 1/(x(10-x)) ?
The integral of 1/(x(10-x)) is 1/10 ln|x|-1/10 ln|x-10|+C