The value of definite integral ∫ limits0(π/2) sin ( sin -1[x]) d x is equal to [Note: [ k ] denotes the largest integer less than or equal to k ]

Question

The value of definite integral ∫ limits0(π/2) sin ( sin -1[x]) d x is equal to [Note: [ k ] denotes the largest integer less than or equal to k ] (A) π-1 (B) (π/2) (C) (π-2/2) (D) (π/4)

Tardigrade Admin · Accepted Answer

Let f(x)= sin ( sin -1[x])= begincases0, 0 ≤ x<1 1, 1 ≤ x<(π/2) endcases ∴ I =∫ limits01 0 dx +∫ limits1(π/2) 1 dx =0+((π/2)-1)=(π-2/2)

Q. The value of definite integral $0 \int \frac{π}{2} sin (sin^{- 1} [x]) d x$ is equal to
[Note: $[k]$ denotes the largest integer less than or equal to $k$ ]

$π - 1$

$\frac{π}{2}$

$\frac{π - 2}{2}$

$\frac{π}{4}$

Let $f (x) = sin (sin^{- 1} [x]) = {0, 1, 0 \leq x < 1 1 \leq x < \frac{π}{2}$
$∴ I = 0 \int 1 0 d x + 1 \int \frac{π}{2} 1 d x = 0 + (\frac{π}{2} - 1) = \frac{π - 2}{2}$

Q. The value of definite integral 0∫2π​​sin(sin−1[x])dx is equal to [Note: [k] denotes the largest integer less than or equal to k ]

π−1

2π​

2π−2​

4π​