`int_(0)^(pi//2)(cosx)/(cos.(x)/2+sin.(x)/2)^(3)dx` का मान ज्ञात कीजिए ।

Explanation

`int_(0)^(pi//2)(cosx)/((cos.(x)/(2)+sin.(x)/(2))^(3))dx=int_(0)^(pi//2)((cos^(2).(x)/(2)-sin^(2).(x)/(2)))/((cos.(x)/(2)+sin.(x)/(2))^(3))dx`
`=int_(0)^(pi//2)(2(cos.(x)/(2)-sin.(x)/(2)))/(2(cos.(x)/(2)+sin.(x)/(2))^(2))dx`
यदि `cos.(x)/(2)+sin.(x)/(2)=t rArr (1)/(2)(cos.(x)/(2)-sin.(x)/(2))dx=dt`
तथा सीमायें हैं `x=0` पर `t=1` तथा `x=pi//2` पर `t=sqrt2,` तब
`int_(0)^(pi//2)(2(cos.(x)/(2)-sin.(x)/(2)))/(2(cos.(x)/(2)+sin.(x)/(2))^(2))dx=2int_(1)^(sqrt2)(dt)/(t^(2))`
`=[(-2)/(t)]^(sqrt2)=sqrt2(sqrt2-1)`