Home
Class 12
MATHS
The value of underset(-pi//2)overset(pi/...

The value of `underset(-pi//2)overset(pi//2)intdx/([x]+[sinx]+4)` where [t] denotes the greatest integer less or equal to t, is

A

`(1)/(12)(7pi-5)`

B

`(1)/(12)(7pi+5)`

C

`(3)/(10) (4 pi-3)`

D

`(3)/(20) (4 pi-3)`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `I= int_(-pi/(2))^(pi/(2))(dx)/([x]+[sinx]+4)`
`= int _(-pi/(2))^(-1)(dx)/([x] + [sinx] +4)+ int_(-1)^(0)(dx)/([x][sinx]+4)`
` + int_(0)^(1)(dx)/([x]+[sinx]+4)+ int_(1)^(pi/2)(dx)/([x]+[sinx]+4)`
and [ sin x] `:' [x]={{:(-2_(,),-pi//2ltxlt-1),(-1_(,),-1ltxlt0),(0_(,),0ltxlt1),(0_(,),1ltxlt pi//2):}`
and `[sinx] = {{:(-1_(,)-pi//2lt x-1,),(-1_(,)-1lt x lt 0,),(0_(,)" "0ltxlt1,),(0_(,)" "1 lt x lt pi//2,):}`
`[:' "For" x lt 0 , - 1le sin x lt 0 and "for" x gt sinx le 1]`
So , `I= int_(-pi/2)^(-1)(dx)/(-2-1+4)+int_(-1)^(0)(dx)/(-1-1+4)+ int_(0)^(1)(dx)/(0+0+4)+ int _(1)^(pi/2)(dx)/(1+0+4)`
`=int_(-pi/(2))^(-1)(dx)/(1)+ int_(-0)^(1) (dx)/(2)+int_(1)^(pi/(2))(dx)/(5)`
`=(-1+(pi)/(2))+(1)/(2)(0+1)+(1)/(4)(1-0)+(1)/(5)((pi)/(2-1))`
`=(-1+(1)/(2)+(1)/(4)-(1)/(5))+((pi)/(2)+(pi)/(10))`
`=(-20+10+5-4)/(20)+(5pi+pi)/(10)`
`=- (9)/(20)+(3pi)/(5)=(3)/(20)(4pi-3)`
Promotional Banner

Similar Questions

Explore conceptually related problems

The value of int_(-pi//2)^(pi//2)dx/([x]+[sinx]+4) where [t] denotes the greatest integer less or equal to t, is

The value of the integral int_(-2)^2 sin^2x/(-2[x/pi]+1/2)dx (where [x] denotes the greatest integer less then or equal to x) is

int_(0)^(pi)[cotx]dx, where [.] denotes the greatest integer function, is equal to

Let f(x) = (x^2-9x+20)/(x-[x]) where [x] denotes greatest integer less than or equal to x ), then

The value of int_(1)^(2)[x]dx , where [x] is the greatest integer less than or equal to x is

The function f(x)=(tan |pi[x-pi]|)/(1+[x]^(2)) , where [x] denotes the greatest integer less than or equal to x, is

let f(x)=(sin4pi[x])/(1+[x]^(2)) , where px] is the greatest integer less than or equal to x then

Solve x^2-4-[x]=0 (where [] denotes the greatest integer function).

Evaluate int_(2)^(5)(x+[x])dx ,where [.] denotes the greatest integer function.

Evaluate int_(2)^(5) (x-[x])dx , where [.] denotes the greatest integer function.