Question

利用定义求定积分定积分号（积分下限0积分上限1）e^x dx

Answer 1

原式=∫(0,1)e^xdx

=lim(n->∞)[e^(1/n)/n+e^(2/n)/n+e^(3/n)/n+.+e^(n/n)/n] (由定积分定义得)

=lim(n->∞){(1/n)[e^(1/n)+e^(2/n)+e^(3/n)+.+e^(n/n)]}

=lim(n->∞){(e^(1/n)/n)[1+e^(1/n)+e^(2/n)+.+e^((n-1)/n)]}

=lim(n->∞){(e^(1/n)/n)[(1-e^(n/n))/(1-e^(1/n))]}

=lim(n->∞){[(1-e)e^(1/n)]*[(1/n)/(1-e^(1/n))]}

=lim(n->∞)[(1-e)e^(1/n)]*lim(n->∞)[(1/n)/(1-e^(1/n))]

=(1-e)*(-1) (∵lim(n->∞)[(1/n)/(1-e^(1/n))]=-1)

=e-1