Question

The value of the integral $\int\limits_{\pi / 3}^{\pi / 2} x \sin (\pi[x]-x) d x$ is (where $[x]$ denotes step up function)

• A. $\frac{1}{2}+\frac{\pi}{6}$
• B. $-\frac{1}{2}-\frac{\pi}{6}$
• C. $1-\frac{\sqrt{3}}{2}+\frac{\pi}{6}$
• D. $\frac{\sqrt{3}}{2}-1-\frac{\pi}{6}$

Answer 1

Answer: (C)

$x \in\left(\frac{\pi}{3}, \frac{\pi}{2}\right) \Rightarrow[x]=1 \quad \Rightarrow I=\int\limits_{\pi / 3}^{\pi / 2} x \sin x d x$ (evaluate using parts)