Solveeit Logo
Login

Question

Question: Evaluate : \(\int _ { - 1 } ^ { 1 } [ x [ 1 + \sin \pi x ] + 1 ]\)dx, where [.] is the greatest inte...

Evaluate : ∫−11[x[1+sin⁡πx]+1]\int _ { - 1 } ^ { 1 } [ x [ 1 + \sin \pi x ] + 1 ]dx, where [.] is the greatest integer function -

A

2

B

1

C

3

D

None of these

Answer

2

Explanation

Solution

Let I = ∫−11[x[1+sin⁡πx]+1]\int _ { - 1 } ^ { 1 } [ x [ 1 + \sin \pi x ] + 1 ] dx,

= ∫−10[x[1+sin⁡πx]+1]\int _ { - 1 } ^ { 0 } [ x [ 1 + \sin \pi x ] + 1 ] dx + ∫01[x[1+sin⁡πx]+1]\int _ { 0 } ^ { 1 } [ x [ 1 + \sin \pi x ] + 1 ] dx

Now [1 + sin px] = 0 if –1 < x < 0

and [1 + sin px] = 1 if 0 < x < 1

\ I =∫−101\int _ { - 1 } ^ { 0 } 1. dx + ∫01[x+1]\int _ { 0 } ^ { 1 } [ x + 1 ] dx

= 1 + 1 ∫01dx\int _ { 0 } ^ { 1 } \mathrm { dx }

= 1 + 1 = 2