The value of the integral ∫ limitsπ / 3π / 2 x sin (π[x]-x) d x is (where [x] denotes step up function)

Question

The value of the integral ∫ limitsπ / 3π / 2 x sin (π[x]-x) d x is (where [x] denotes step up function) (A) (1/2)+(π/6) (B) -(1/2)-(π/6) (C) 1-(√3/2)+(π/6) (D) (√3/2)-1-(π/6)

Tardigrade Admin · Accepted Answer

x ∈((π/3), (π/2)) ⇒[x]=1 ⇒ I=∫ limitsπ / 3π / 2 x sin x d x (evaluate using parts)

Q. The value of the integral $π /3 \int π /2 x sin (π [x] - x) d x$ is (where $[x]$ denotes step up function)

$\frac{1}{2} + \frac{π}{6}$

$- \frac{1}{2} - \frac{π}{6}$

$1 - \frac{3}{2} + \frac{π}{6}$

$\frac{3}{2} - 1 - \frac{π}{6}$

$x \in (\frac{π}{3}, \frac{π}{2}) \Rightarrow [x] = 1 \Rightarrow I = π /3 \int π /2 x sin x d x$ (evaluate using parts)

Q. The value of the integral π/3∫π/2​xsin(π[x]−x)dx is (where [x] denotes step up function)

21​+6π​

−21​−6π​

1−23​​+6π​

23​​−1−6π​