What is the integral from 1 to 4 of ln(2x)-ln(x) ?

The integral from 1 to 4 of ln(2x)-ln(x) is 3ln(2)

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integral from 1 to 4 of ln(2x)-ln(x)

\int_{1}^{4} ln (2 x) - ln (x) d x

3 ln (2)

Decimal

2.07944 \dots

Solution steps

\int_{1}^{4} 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}^{4} ln (2 x) d x - \int_{1}^{4} ln (x) d x

\int_{1}^{4} ln (2 x) d x = 11 ln (2) - 3

\int_{1}^{4} ln (x) d x = 8 ln (2) - 3

= 11 ln (2) - 3 - (8 ln (2) - 3)

Simplify

11 ln (2) - 3 - (8 ln (2) - 3) : 3 ln (2)

= 3 ln (2)

\int_{0}^{2} x 4 + x^{2} d x

\int_{- 3}^{3} e^{x + 3} d x

\int_{0}^{2} - t^{2} + 3 t - 2 d t

\int_{0}^{4} (x^{2} - 3) d x

\int_{2}^{5} 4^{4 x} d x

What is the integral from 1 to 4 of ln(2x)-ln(x) ?
The integral from 1 to 4 of ln(2x)-ln(x) is 3ln(2)