What is the integral from 1 to 2 of ln(2x)-ln(x) ?

The integral from 1 to 2 of ln(2x)-ln(x) is ln(2)

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integral from 1 to 2 of ln(2x)-ln(x)

\int_{1}^{2} ln (2 x) - ln (x) d x

ln (2)

Decimal

0.69314 \dots

Solution steps

\int_{1}^{2} 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_{1}^{2} ln (2 x) d x - \int_{1}^{2} ln (x) d x

\int_{1}^{2} ln (2 x) d x = 3 ln (2) - 1

\int_{1}^{2} ln (x) d x = 2 ln (2) - 1

= 3 ln (2) - 1 - (2 ln (2) - 1)

Simplify

3 ln (2) - 1 - (2 ln (2) - 1) : ln (2)

= ln (2)

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What is the integral from 1 to 2 of ln(2x)-ln(x) ?
The integral from 1 to 2 of ln(2x)-ln(x) is ln(2)