计算∫（上积分1,下积分0）e^(x^1/2)

题目

计算∫（上积分1,下积分0）e^(x^1/2)

答案

令x^(1/2)=t (t≥0),则x=t²
0≤x≤1,0≤t≤1
∫(0 1)e^(x^1/2)dx
=∫(0 1)e^t d(t²)
=∫(0 1)2te^t dt
=2te^t-2e^t|(0 1)
=(2×1×e^1 -2×e^1)-(2×0×e^0 -2×e^0)
=0-0+2
=2