What is the integral of ln(2x)-ln(x) ?

The integral of ln(2x)-ln(x) is xln(2x)-xln(x)+C

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integral of ln(2x)-ln(x)

\int ln (2 x) - ln (x) d x

x ln (2 x) - x ln (x) + C

Solution steps

\int ln (2 x) - ln (x) 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 (2 x) d x - \int ln (x) d x

\int ln (2 x) d x = x ln (2 x) - x

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

= x ln (2 x) - x - (x ln (x) - x)

Simplify

= x ln (2 x) - x ln (x)

Add a constant to the solution

= x ln (2 x) - x ln (x) + C

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What is the integral of ln(2x)-ln(x) ?
The integral of ln(2x)-ln(x) is xln(2x)-xln(x)+C