Definite Integration
Integral Calculus-2
star_batch_jee_advanced_2025
Grade 12
Question:
The value of $\int_1^8 x\sin[x^2 - \pi] dx$, where $[.]$ denotes the greatest integer function is:
$\sum_{r=1}^6 r\sin r
$\sum_{r=1}^6 (-1)^r r\sin r
$\sum_{r=1}^6 r^2 \sin r
None of these
Step-by-Step Solution
Key Concept: Odd and even function symmetries, combined with periodicity properties of trigonometric functions, simplify definite integrals.
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