Question

求广义积分∫xe^(-x^2)dx,其中积分上限是+∞，积分下限是1，求详细过程~

Answer 1

∫(0→∞)xe^(-x²)dx
=½∫(0→∞)e^(-x²)dx²
=e^(-x²)(0→∞)
=-½(0-1)
=
½
∫(0→1)lnx
dx
=xlnx(0→1)
-
∫(0→1)x(1/x)dx
[分部积分]
=1ln1
-
xlnx
(x→0)
-
∫(0→1)dx
=
0
-
xlnx
(x→0)
-
x(0→1)
=-xlnx(x→0)
-
1
=
-
(lnx)/(1/x)(x→0)
-
1
=
-(1/x)/(-1/x穿担扁杆壮访憋诗铂涧²)(x→0)
-
1
[运用了罗毕达方法]
=
x(x→0)
-
1
=
0
-
1
=
-1