निम्नलिखित को सिद्ध कीजिए ।
`int_(0)^((pi)/(2))sin^(3)x dx =(2)/(3)`

Explanation

माना `I=int_(0)^(pi//2)sin^(3)x dx`
`" "=int_(0)^(pi//2)sin^(2)x. sinx dx`
`" "=int_(0)^(pi//2)(1-cos^(2)x)sinxdx`
माना `cosx=t`
`rArr" "-sinx dx=dt`
`rArr" "sinx dx = 0dt`
तथा`" "x=0" "rArr" "t=cos 0 =1`
और `" "x=(pi)/(2)" "rArr" "t=cos.(pi)/(2)=0`
`therefore" "I=int_(0)^(pi//2)(1-cos^(2)x)sinx dx`
`" "=int_(1)^(0)(1-t^(2))(-dt)=-[t-(t^(3))/(3)]_(1)^(0)`
`" "=-{(0-0)-(1-(1)/(3))}=(2)/(3)`