What is the integral of ln(((9+x))/x) ?

The integral of ln(((9+x))/x) is xln((9+x)/x)+9ln|x+9|+C

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integral of ln(((9+x))/x)

integral

ln (\frac{( 9 + x )}{x})

x ln (\frac{9 + x}{x}) + 9 ln ∣ x + 9 ∣ + C

Solution steps

\int ln (\frac{9 + x}{x}) d x

Apply Integration By Parts

= x ln (\frac{9 + x}{x}) - \int - \frac{9}{x + 9} d x

\int - \frac{9}{x + 9} d x = - 9 ln ∣ x + 9 ∣

= x ln (\frac{9 + x}{x}) - (- 9 ln ∣ x + 9 ∣)

Simplify

= x ln (\frac{9 + x}{x}) + 9 ln ∣ x + 9 ∣

Add a constant to the solution

= x ln (\frac{9 + x}{x}) + 9 ln ∣ x + 9 ∣ + C

What is the integral of ln(((9+x))/x) ?
The integral of ln(((9+x))/x) is xln((9+x)/x)+9ln|x+9|+C