A teacher tells her class that some integrals hide a symmetry…

A teacher tells her class that some integrals hide a symmetry. She gives the integral $I = \displaystyle\int_0^{\pi/2}\dfrac{\sin x}{\sin x + \cos x}\,dx$ and says that it cannot be done with a simple antiderivative. A student notices that $\cos x = \sin\left(\dfrac{\pi}{2} - x\right)$, and uses the property $\displaystyle\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$ with $a = \dfrac{\pi}{2}$. This gives a second integral $J = \displaystyle\int_0^{\pi/2}\dfrac{\cos x}{\sin x + \cos x}\,dx$, which equals $I$. She adds the two integrals and finds that the integrands add up to a constant. The teacher then asks for one more integral, $K = \displaystyle\int_0^{\pi/2}\dfrac{\sin x + 2\cos x}{\sin x + \cos x}\,dx$, which can be written with $I$ and $J$. The class checks each answer by thin strips on a computer, but the exact value must come from a short argument. She wants each step to be easy to follow for the rest of the class. Find $I$, then $J$, then $K$. Give each to four decimal places.