Find $\int_0^{\frac{\pi}{2}} e^{-a(\sin(x)+\cos(x))} \, dx$ and/or $\int_0^{\frac{\pi}{2}} e^{-a(\sin(x)+\cos(x))} \sin(\cos(x)) \, dx$

Question

While trying to solve a certain integral I was left with these rather silly integral.

$$\int_0^{\frac{\pi}{2}} e^{-a(\sin(x)+\cos(x))} \, dx$$ $$\int_0^{\frac{\pi}{2}} e^{-a(\sin(x)+\cos(x))} \sin(\cos(x)) \, dx$$

Any headway on these problems would be greatly appreciated!