大学微积分(30题)
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I(m,n)=1/(n+1)*∫(0,1) (lnx)^m d[x^(n+1)]
=1/(n+1)*(lnx)^m*x^(n+1)|(0,1)-m/(n+1)*∫(0,1) x^n*(lnx)^(m-1) dx
=-1/(n+1)*lim(x->0+) (lnx)^m*x^(n+1)-m/(n+1)*I(m-1,n)
=-m/(n+1)*I(m-1,n)
=[-m/(n+1)]*[-(m-1)/(n+1)]*I(m-2,n)
......
=m!*[-1/(n+1)]^m*I(0,n)
=m!*[-1/(n+1)]^m*∫(0,1) x^n dx
=m!*[-1/(n+1)]^m*[1/(n+1)]*x^(n+1)|(0,1)
=(-1)^m*m!/(n+1)^(m+1)
=1/(n+1)*(lnx)^m*x^(n+1)|(0,1)-m/(n+1)*∫(0,1) x^n*(lnx)^(m-1) dx
=-1/(n+1)*lim(x->0+) (lnx)^m*x^(n+1)-m/(n+1)*I(m-1,n)
=-m/(n+1)*I(m-1,n)
=[-m/(n+1)]*[-(m-1)/(n+1)]*I(m-2,n)
......
=m!*[-1/(n+1)]^m*I(0,n)
=m!*[-1/(n+1)]^m*∫(0,1) x^n dx
=m!*[-1/(n+1)]^m*[1/(n+1)]*x^(n+1)|(0,1)
=(-1)^m*m!/(n+1)^(m+1)
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