Integral #1

$\begin{aligned} f(x) & = \int e^x \sin x \cos x \ dx \\ & = \frac 12 \int e^x \sin 2x \ dx & \small \blue{\text{By integration by parts}} \\ & = \frac 12 \left(e^x \sin 2x - \int 2 e^x \cos 2x \ dx \right) \\ & = \frac 12 e^x \sin 2x - e^x \cos 2x - 2 \blue{\int e^x \sin 2x \ dx} & \small \blue{\text{Note that }f(x) = \frac 12 \int e^x \sin 2x \ dx} \\ & = \frac 12 e^x \sin 2x - e^x \cos 2x - 4 \blue{f(x)} \\ & = \frac {e^x}{10} (\sin 2x - 2\cos 2x) + \blue C & \small \blue{\text{where }C \text{ is the constant of integration.}} \\ f(0) & = \frac {-2}{10} + C = 3 & \small \blue{\implies C = 3.2} \\ f \left(\frac \pi 8 \right) & = \frac {e^\frac \pi 8}{10}\left(\frac 1{\sqrt 2} - \sqrt 2\right) + 3.2 \end{aligned}$

Therefore, $a+b+c = 3.2 + \frac 1{10}\left(\frac 1{\sqrt 2} - \sqrt 2\right) + \frac \pi 8 \approx \boxed{3.52}$ .