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The Integral int(pi/4)^((3pi)/4)(dx)/(1+...

The Integral `int_(pi/4)^((3pi)/4)(dx)/(1+cosx)` is equal to: (2) (3) (4)

A

2

B

`-2`

C

`(1)/(2)`

D

`-(1)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `I=int_(pi//4)^(3pi//4)(dx)/(1+cosx)` . . . (i)
`rArrI=int_(pi//4)^(3pi//4)(dx)/(1+cos(pi-x))`
`I=int_(pi//4)^(3pi//4)(dx)/(1+cosx)` . . . (ii)
On adding Eqs . (i) and (ii) , we get
`2I=int_(pi//4)^(3pi//4)((1)/(1+cosx)+(1)/(1-cosx))`dx
`rArr2I=int_(pi//4)^(3pi//4)((2)/(1-cos^(2)x))`dx
`rArrI=int_(pi//4)^(3pi//4)"cosec"^(2)xdx=[-cotx]_(pi//4)^(3pi//4)`
`=[-cot(3pi)/(4)+cot(pi)/(4)]=-(-1)+1=2`
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