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-》00
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![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-》00](/uploads/image/z/10379390-14-0.jpg?t=lim+%281%2Fx%5E2%29%2A%E2%88%AB%5B%281%2B2t%29%2Ae%5E%28t-x%5E2%29%5Ddt%3D%3F%E6%B3%A8%EF%BC%9A%E7%A7%AF%E5%88%86%E8%8C%83%E5%9B%B4%E6%98%AF0-%E3%80%8Bx%5E2+lim%E7%9A%84%E4%B8%8B%E9%9D%A2%E6%98%AFx-%E3%80%8B00)
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-》00
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-》00
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-》00
=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