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Evaluate : (i) int(0)^(pi//4)sqrt(1+sin2...

Evaluate : `(i) int_(0)^(pi//4)sqrt(1+sin2x)\ dx` `(ii) int_(0)^(pi//2)sqrt(1+cos2x)\ dx`

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`(i) int_(0)^(pi//4)sqrt(1+sin2x)dx=int_(0)^(pi//4)sqrt(cos^(2)x+sin^(2)x+2sinxcosx)dx`
`=int_(0)^(pi//4)(cosx+sinx)dx=[sinx-cosx]_(0)^(pi//4)=1`.
`(ii) int_(0)^(pi//2)sqrt(1+cos2x)dx=int_(0)^(pi//2)sqrt(2cos^(2)x)dx`
`=sqrt(2)int_(0)^(pi//2)cosxdx=sqrt(2)[sinx]_(0)^(pi//2)=sqrt(2)`.
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