`int_(0)^((pi)/(2))(sin^((3)/(2))x)/(sin^((3)/(2))+cos^((3)/(2))x)dx" ...(1)"`

Explanation

माना `I= int_(0)^((pi)/(2))(sin^((3)/(2))x)/(sin^((3)/(2))+cos^((3)/(2))x)dx`
`rArr" "I=int_(0)^(pi//2)(sin^((3)/(2))((pi)/(2)-x))/(sin^((3)/(2))((pi)/(2)-x)+cos^((3)/(2))((pi)/(2)-x))dx`
`" "[because int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx]`
`" "=int_(0)^(pi//2)(cos^((3)/(2)))/(cos^((3)/(2))x+sin^((3)/(2))x)dx" ...(2)"`
समीकरण (1 ) और (2 ) को जोड़ने पर ,
`2I=int_(0)^(pi//2)(sin^((3)/(2))+cos^((3)/(2))x)/(sin^((3)/(2))x+cos^((3)/(2))x)dx`
`=int_(0)^(pi//2)1dx=[x]_(0)^(pi//2)=(pi)/(2)-0`
`rArr" "I=(pi)/(4)`