Ta có:
$\int\limits_0^{\frac{\pi}{2}}(e^{\sin x}+\cos x)\cos xdx$
$=\int\limits_0^{\frac{\pi}{2}}e^{\sin x}\cos xdx+\int\limits_0^{\frac{\pi}{2}}\cos^2xdx$
$=\int\limits_0^{\frac{\pi}{2}}e^{\sin x}d(\sin x)+\dfrac{1}{2}\int\limits_0^{\frac{\pi}{2}}(1+\cos2x)dx$
$=\left(e^{\sin x}+\dfrac{x}{2}+\dfrac{\sin2x}{4}\right)\left|\begin{array}{l}\dfrac{\pi}{2}\\0\end{array}\right.$
$=e-1+\dfrac{\pi}{4}$