计算积分∫0→θ x^2/θ(1-x/θ)^(n-1)dx

计算积分∫0→θ x^2/θ(1-x/θ)^(n-1)dx
计算积分∫0→θ x^2/θ(1-x/θ)^(n-1)dx

计算积分∫0→θ x^2/θ(1-x/θ)^(n-1)dx
∫0→θ x^2/θ(1-x/θ)^(n-1)dx
=θ^2*∫0→1( x^2(1-x)^(n-1)dx)
=θ^2*∫0→1(-1/n*x^2*d((1-x)^n))
=θ^2/n*[-x^2(1-x)^n|(0->1)+∫0→1(2x(1-x)^ndx)]
=2θ^2/(n(n+1))*∫0→1(-xd((1-x)^(n+1)))
=2θ^2/(n(n+1))*[-x(1-x)^(n+1)|(0->1)+∫0→1((1-x)^(n+1)dx)
=2θ^2/(n(n+1)(n+2))