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Show that`int_0^(pi/2)sqrt((sin2theta))sinthetadtheta=pi/4` |
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Answer» Let `I=int_(0)^(pi//2)sqrt((sin 2theta))sin theta d theta `……………1 `:.I=int_(0)^(pi//2)sqrt(sin(2(pi/2)-theta))sin(pi/2-theta)d theta` `=int_(0)^(pi//2)sqrt((sin 2theta))cos theta d theta` …………..2 Adding 1 and 2 we get `2I=int_(0)^(pi//2)sqrt((sin 2theta))(sin theta+costheta)d theta` or `I=1/2int_(0)^(pi//2)sqrt(1-(sin theta -cos theta)^(2))(sin theta+cos theta) d theta` `=1/2int_(-1)^(1)sqrt(1-t^(2))dt`[ Let `sin theta -cos theta=t`] `=1/2[1/2 t sqrt((1-t^(2)))+1/2sin^(-1)t]_(-1)^(1)=(pi)/4` |
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