First slide

Evaluation of definite integrals

Question

The value of $\int_{0}^{1} max (e^{x}, e^{1 - x}) d x$ equals

Moderate

A

$2 (e - 1)$
B

$2 (e - \sqrt{e})$
C

$2 (e + \sqrt{e})$
D

$2 (e + 1)$

Solution

$\begin{array}{l} \int_{0}^{1} max (e^{x}, e^{1 - x}) d x \\ = \int_{0}^{1 / 2} max (e^{x}, e^{1 - x}) d x + \int_{1 / 2}^{1} max (e^{x}, e^{1 - x}) d x \\ = \int_{0}^{1 / 2} e^{1 - x} d x + \int_{1 / 2}^{1} e^{x} d x \\ = - {e^{1 - x}|}_{0}^{1 / 2} + e - e^{1 / 2} \\ = - e^{1 / 2} + e + e - e^{1 / 2} \\ = 2 (e - e^{1 / 2}) \end{array}$

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