a: \(\int\left(2\cdot cosx+sinx\right)dx\)
\(=2\cdot sinx-cosx+C\)
\(\int_0^{\dfrac{\Omega}{4}}\left(2\cdot cosx+sinx\right)dx\)
\(=\left[2\cdot sin\left(\dfrac{\Omega}{4}\right)-cos\left(\dfrac{\Omega}{4}\right)+C\right]-\left[2\cdot sin0-cos0+C\right]\)
\(=\left[2\cdot\dfrac{\sqrt{2}}{2}-\dfrac{\sqrt{2}}{2}+C\right]-\left[2\cdot0-1\right]\)
\(=\dfrac{\sqrt{2}}{2}+1=\dfrac{2+\sqrt{2}}{2}\)
b: \(\int\left(sinx-cosx\right)dx=-cosx-sinx+C\)
\(\int_0^{\dfrac{\Omega}{3}}\left(sinx-cosx\right)dx\)
\(=\left(-cos\left(\dfrac{\Omega}{3}\right)-sin\left(\dfrac{\Omega}{3}\right)+C\right)-\left(-cos0-sin0+C\right)\)
\(=\left(-\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}+C\right)-\left(-1-0+C\right)\)
\(=\dfrac{-1-\sqrt{3}}{2}+1=\dfrac{-1-\sqrt{3}+2}{2}=\dfrac{1-\sqrt{3}}{2}\)
