Answer
$$\int_0^1\cos(\pi t/2)dt=\frac{2}{\pi}$$
Work Step by Step
To evaluate the integral $$\int_0^1\cos(\pi t/2)dt$$ we will use substitution $\pi t/2=z$ which gives us $\pi/2dt=dz\Rightarrow dt=2/\pi dx$ and now integration bounds would be: for $t=0$ we have $z=0$ and for $t=1$ we have $z=\pi/2.$ Now, putting all this into the integral we get:
$$\int_0^1\cos(\pi t/2)dt=\int_0^{\pi/2}\cos z\cdot2/\pi dz=
\frac{2}{\pi}\left.\sin z\right|_0^{\pi/2}=\frac{2}{\pi}(\sin\frac{\pi}{2}-\sin0)=\frac{2}{\pi}(1-0)=\frac{2}{\pi}$$
