Question

Find the average value of f(x) = 1/x over the interval [e, 2e].

Answer 1

$\bf \displaystyle \cfrac{1}{2e-e}\int_{e}^{2e}~\cfrac{1}{x}\cdot dx\implies \left. \cfrac{1}{e}\cdot ln(x) \right]_{e}^{2e}\implies \left[ \cfrac{ln(2e)}{e} \right]-\left[ \cfrac{ln(e)}{e}\right] \\\\\\ \left[ \cfrac{ln(2)+ln(e)}{2e} \right]-\left[ \cfrac{1}{e} \right]\implies \cfrac{ln(2)}{e}+\cfrac{ln(e)}{e}-\cfrac{1}{e}\implies \cfrac{ln(2)}{e}+\cfrac{1}{e}-\cfrac{1}{e} \\\\\\ \cfrac{ln(2)}{e}$