设f(x)=∫(0,1-x)e^t(2-t)dt,求I=∫(0,1)f(x)dx

问题描述:

设f(x)=∫(0,1-x)e^t(2-t)dt,求I=∫(0,1)f(x)dx

f(x) = ∫e^[t(2-t)]dt,
f'(x) = -e^[(1-x)(1+x)] = -e^(1-x^2) = -ee^(-x^2)
I = ∫f(x)dx= -e ∫e^(-x^2)dx 收敛但积不出来

若是求 I = ∫f(x)dx , 则
I = ∫f(x)dx= -e ∫e^(-x^2)dx= -(1/2)e √π