What is the integral of (ln(x)-ln(xy)) ?

The integral of (ln(x)-ln(xy)) is -xln(yx)+xln(x)+C

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integral of (ln(x)-ln(xy))

\int (ln (x) - ln (x y)) d x

- x ln (y x) + x ln (x) + C

Solution steps

\int ln (x) - ln (x y) 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 ln (y x) d x + \int ln (x) d x

\int ln (y x) d x = x ln (y x) - x

\int ln (x) d x = x ln (x) - x

= - (x ln (y x) - x) + x ln (x) - x

Simplify

= - x ln (y x) + x ln (x)

Add a constant to the solution

= - x ln (y x) + x ln (x) + C

x \to \infty lim (i x)

\frac{d}{d x} (4 x^{3} - 5 x^{2} + 4 x - 1)

x \to - \infty lim (\frac{19}{20 + ( \frac{1}{x} )})

\int_{2}^{4} x e^{\frac{x}{2}} d x

f (x) = \frac{1}{2 x - 1}

What is the integral of (ln(x)-ln(xy)) ?
The integral of (ln(x)-ln(xy)) is -xln(yx)+xln(x)+C