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

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

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

integral

ln (\frac{x}{( 10 - x )})

x ln (\frac{x}{10 - x}) + 10 ln ∣ - x + 10 ∣ + C

Solution steps

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

Apply Integration By Parts

: x ln (\frac{x}{10 - x}) - \int \frac{10}{- x + 10} d x

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

\int \frac{10}{- x + 10} d x = - 10 ln ∣ - x + 10 ∣

= x ln (\frac{x}{10 - x}) - (- 10 ln ∣ - x + 10 ∣)

Simplify

= x ln (\frac{x}{10 - x}) + 10 ln ∣ - x + 10 ∣

Add a constant to the solution

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

\frac{( x - 3 )}{( x ^{2} - 3 x + 2 )}

\frac{x ^{3}}{x ^{4} + 1}

e^{3 x - 1}

e^{2}

e^{3 x - 2}

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