MHT CET20244 May 2024Morning ShiftMathematicsDefinite IntegrationActual
The integral _ -1 2 ^ 1 2 ([x]+ _c ( 1+x 1-x ) ) d x , where [x] represent greatest integer function, equals
Options
- A- 1 2
- B_e ( 1 2 )
- C1 2
- D2 _ e ( 1 2 )
Correct answer
A. - 1 2
Step-by-step solution
Let I = _ -1 2 ^ 1 2 ([x]+ _ e ( 1+x 1-x ) ) d x= _ -1 2 ^0[x] d x+ ₀^ 1 2 [x] d x+ _ -1 2 ^ 1 2 _ e ( 1+x 1-x ) d x Let g(x)= ( 1+x 1-x )g(-x)= ( 1-x 1+x )=- ( 1+x 1-x )=-g(x) g (x) is a odd function. _ -1 2 ^ 1 2 ~g (x) d x=0 aligned I & = _ -1 2 ^0(-1) d x+ ₀^ 1 2 (0) d x+0 & =[-x]_ -1 2 ^0+0 & = -1 2 aligned