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int (0)^(pi//3) (cos x + sin x)/(sqrt(1+...

` int _(0)^(pi//3) (cos x + sin x)/(sqrt(1+sin 2x))dx ` is equal to

A

`(4pi)/(3)`

B

`(2pi)/3`

C

`pi`

D

`pi/3`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `l=int_(0)^(pi//3)(cosx+sinx)/(sqrt(1+2sinxcosx))dx`
`=int_(0)^(pi//3)(cosx+sinx)/(sqrt(cos^(2)x+sin^(2)x+2sinxcosx))dx`
`=int_(0)^(pi//3)(cosx+sinx)/(sqrt((cosx+sinx)^(2)))dx`
`=int_(0)^(pi//3)(cosx+sinx)/(cosx+sinx)dx=int_(0)^(pi//3)1dx=(pi)/(3)`
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