Answer
$\int_{0}^{1}e^x~cos~x~dx \leq e-1$
Work Step by Step
Note that $e^x \gt 0$ for all $x$
On the interval $[0,1]$:
$cos~x \leq 1$
$e^x~cos~x \leq e^x$
Therefore:
$\int_{0}^{1}e^x~cos~x~dx \leq \int_{0}^{1}e^x~dx$
$\int_{0}^{1}e^x~cos~x~dx \leq e^x~\Big\vert_{0}^{1}$
$\int_{0}^{1}e^x~cos~x~dx \leq e-1$
