First slide

Evaluation of definite integrals

Question

The mean value of the function $f (x) = \frac{2}{e^{x} + 1}$ on the interval [0, 2] is

Moderate

A

$\log \frac{2}{e^{2} + 1}$
B

$1 + \log \frac{2}{e^{2} + 1}$
C

$2 + \log \frac{2}{e^{2} + 1}$
D

$2 + \log (e^{2} + 1)$

Solution

Average = $\frac{1}{2} \int_{0}^{2} \frac{2}{e^{x} + 1} d x = \int_{0}^{2} \frac{e^{- x}}{1 + e^{- x}} d x$
$\begin{array}{l} {= - \log (1 + e^{- x})]}_{0}^{2} \\ = \log 2 - \log (1 + e^{- 2}) \\ = 2 + \log [2 / (1 + e^{2})] \end{array}$

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