`int_(0)^((pi)/(2))(cos^(5)xdx)/(sin^(5)x+cos^(5)x)`

Explanation

माना `I=int_(0)^(pi//2)(cos^(5)x)/(sin^(5)x+cos^(5)x)dx" …(1)"`
`rArr" "I=int_(0)^(pi//2)(cos^(5)((pi)/(2)-x))/(sin^(5)((pi)/(2)-x)+cos^(5)((pi)/(2)-x))dx`
`[because int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx]`
`=int_(0)^(pi//2)(sin^(5)x)/(cos^(5)x+sin^(5)x)dx" ...(2)"`
समीकरण (1 ) और (2 ) को जोड़ने पर ,
`2I=int_(0)^(pi//2)(sin^(5)x+cos^(5)x)/(sin^(5)x+cos^(5)x)dx`
`=int_(0)^(pi//2)1.dx = [x]_(0)^(pi//2)=(pi)/(2)`
`rArr" "I=(pi)/(4)`