What is the integral from 1 to 3 of ln(3x)-ln(x) ?

The integral from 1 to 3 of ln(3x)-ln(x) is 2ln(3)

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integral from 1 to 3 of ln(3x)-ln(x)

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

2 ln (3)

Decimal

2.19722 \dots

Solution steps

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

\int_{1}^{3} ln (3 x) d x = 5 ln (3) - 2

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

= 5 ln (3) - 2 - (3 ln (3) - 2)

Simplify

5 ln (3) - 2 - (3 ln (3) - 2) : 2 ln (3)

= 2 ln (3)

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What is the integral from 1 to 3 of ln(3x)-ln(x) ?
The integral from 1 to 3 of ln(3x)-ln(x) is 2ln(3)