Step 1: Quick sanity check:
The integrand is positive and small near $0$ and about $\frac{5\times 2}{8} = 1.25$ at the upper limit, over an interval of length about $1.61$, so the integral should be of order $1$.
Step 2: Compute:
$4 - \pi \approx 0.858$. The options $5 \pm \pi$ and $4 + \pi$ are far larger, and the careful substitution gives exactly $4 - \pi$.
Final Answer:
The integral equals $4 - \pi$, option (D).
\[ \boxed{4-\pi} \]