lim (1/x^2)*∫[(1+2t)*e^(t-x^2)]dt=?注：积分范围是0-》x^2 lim的下面是x-》0

=lim (1/x^2)*∫[(1+2(t-x^2 + x^2))*e^(t-x^2)]d(t-x^2)
令u=t-x^2,则u的范围是 -x^2 至 0.
原式=lim (1/x^2)*∫[(1+2(u + x^2))*e^u ]du
=lim (1/x^2)*{∫[(1 + 2x^2 + 2u )*e^u ]du }
=lim (1/x^2)*{∫(1 + 2x^2)e^u du + 2∫u *e^u du }
=lim {(1 + 2x^2)∫e^u du + 2∫u *e^u du } /x^2
=lim {(1 + 2x^2)e^u|< u从-x^2 至 0> + 2∫u *e^u du } /x^2
=lim {(1 + 2x^2)(1-e^(-x²) + 2∫u *e^u du ) } /x^2
=lim [(1 + 2x^2)/x^2]·lim (1-e^(-x²) + 2∫u *e^u du )
= 2·lim (1-0 + 2∫u *e^u du )
= 2·lim (1 + 2∫u de^u )
= 2·lim (1 + 2(u·e^u -∫e^u du ) )
= 2·lim (1 + 2(u-1)·e^u| )
= 2·lim (1 + 2[(0-1)·e^0) - (-x^2 -1)·e^(-x^2) ] )
= 2·lim (1 + 2[-1 + (x^2 +1)·e^(-x^2) ] )
= 2·lim (1 + 2[-1 + (x^2 +1)·e^(-x^2) ] )
= -2 + 4·lim (x^2 +1) / e^(x^2)
= -2 + 4·lim (2x) / [2x·e^(x^2)] 【洛比达法则】
= -2 + 4·lim 1 / e^(x^2)
= -2 + 0
= -2