Question

Find the average value of the function g(x) = lnx/x over the interval [1, e].

Answer 1

`\displaystyle \bar g=(1)/(2(e-1))`

Explanation:

Average Value of a Function

Given a function g(x), the average value of g in a given interval (xo,x1) is given by

`\displaystyle \bar g=(1)/(x_1-x_0)\int_(x_0)^(x_1) g(x)dx`

Plugging in the given data

`\displaystyle \bar g=(1)/(e-1)\int_(1)^(e) (lnx)/(x)dx`

Let's compute the indefinite integral

`\displaystyle I=\int (lnx)/(x)dx`

We'll use the substitution u=lnx, du=dx/x. Then

`\displaystyle I=\int u.du`

`\displaystyle I=(u^2)/(2)`

Taking back the substitution

`\displaystyle I=(ln^2x)/(2)`

The average value is

`\displaystyle \bar g=(1)/(e-1)(ln^2x)/(2) \ | _1^e`

`\displaystyle \bar g=(1)/(e-1)\left((ln^2e)/(2)-(ln^21)/(2) \right )`

`\displaystyle \bar g=(1)/(e-1)\left((1)/(2)-0 \right )`

`\displaystyle \bar g=(1)/(2(e-1))`