Home
Class 12
MATHS
Prove that int0^oo[cot^(-1)x]dx ,w h e r...

Prove that `int_0^oo[cot^(-1)x]dx ,w h e r e[dot]` denotes the greatest integer function.

Text Solution

Verified by Experts

The correct Answer is:
`pi+cot1+cot2`
We have `int_(-pi//2)^(2pi)[cot^(-1)x]dx`
We have that `cot^(-1)xepsilon[0,pi]`
so `[cot^(-1)x]=0` for `cot^(-1)x epsilon(0,1)` or `xepsilon(cot1,oo)`
`[cot^(-1)x]=1` for `cot^(-1)xepsilon[1,2)` or `xepsilon(cot2, cot1]`
`[cot^(-1)x=2` for `cot^(-1)x epsilon[2,3)` or `x epsilon(cot3, cot2]`
`[cot^(-1)x]=3` for `cot^(-1)xepsilon[3,pi)` or `x epsilon(-oo,cot3]`
`:. int_(cot1)^(2pi) 0dx+int_(cot2)^(cot1) 1dx+int_(-pi//2)^(cot2) 2dx`
`=cot1-cot2+2cot2+pi`
`=cot1+cot2+pi`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.5|11 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.6|7 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.3|4 Videos

  • CURVE TRACING

    CENGAGE|Exercise Exercise|24 Videos

  • DETERMINANT

    CENGAGE|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Prove that int_(-pi/2)^(2pi)[cot^(-1)x]dx ,where [dot] denotes the greatest integer function.

Evaluate: int_0^2[x^2-x+1]dx ,w h e r e[dot] denotos the greatest integer function.

The integral int_0^(1. 5)[x^2]dx ,w h e r e[dot] denotoes the greatest integer function, equals ...........

Evaluate: int_0^(2pi)[sinx]dx ,w h e r e[x] denotes the greatest integer function.

Evaluate: int_1^(e^6)[(logx)/3]dx ,w h e r e[dot] denotes the greatest integer function.

Evaluate: int_0^oo[2e^(-x)]dx ,w h e r e[x] represents greatest integer function.

Evaluate: int_(-5)^5x^2[x+1/2]dx(w h e r e[dot] denotes the greatest integer function).

Evaluate int_(-1)^(3) [x]dx ,where [.] denotes the greatest integer function.

Evaluate: int_0^((5pi)/(12))[tanx]dx , where [dot] denotes the greatest integer function.

Evaluate: int_0^(100)x-[x]dx where [dot] represents the greatest integer function).