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int (0)^(pi //2)((root(n)(secx))/(root(n...

`int _(0)^(pi //2)((root(n)(secx))/(root(n)(secx)+root(n)("cosec"x)))dx=`

A

`pi/2`

B

`pi/3`

C

`pi/4`

D

`pi/6`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `l=int_(0)^((pi)/(2))(root(n)(sec x))/(root(n)(sec x)+root(n)("cosec" x))dx` …(i)
`= int_(0)^((pi)/(2))(root(n)(sec((pi)/(2)-x)))/(root(n)(sec((pi)/(2)-x))+root(n)("cosec "((pi)/(2)-x)))dx`
`= int_(0)^((pi)/(2))(root(n)("cosec" x))/(root(n)("cosec" x)+root(n)(sec x))dx` ...(ii)
On adding Eq. (i) and (ii), we get
`2l+int_(0)^((pi)/(2))(root(n)(sec x)+root(n)("cosec " x))/(root(n)(sec x)+root(n)("cosec "x))dx`
`rArr 2l = int_(0)^((pi)/(2))dx=[x]_(0)^(pi//2)`
`rArr 2l=(pi)/(2) rArr l = (pi)/(4)`
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