The value of $\int_1^8 x\sin[x^2 - \pi] dx$, where $[.]$ denotes the greatest integer function is:
Step-by-Step Solution
Key Concept: The greatest integer function [x² - π] creates piecewise constant intervals on [1,8]. Since π ≈ 3.14, we have x² - π ranging from -2.14 to 61.86, so [x² - π] takes integer values 0,1,2,...,8 on disjoint intervals. The integral splits as Σ∫ r·x·sin(r)dx over intervals where [x² - π] = r, but evaluating these requires solving x² = r + π for bounds.
The integral $\int_{-π}^π x\sin(x^2-π)dx = 2\int_0^π x\sin(x^2-π)dx$ by symmetry considerations. Since $-π ≤ x^2 - π ≤ π^2 - π ≤ π^2 - 7$ and $-π ≤ x^2 - π$ for all $x$ in the domain, the last integral equals $2\int_0^π x\sin(x^2)dx$ by periodicity of sine.
Correct Answer: 4