Question

∫[上限为1,下限为0] (e^x2)(x-x^3)dx

Answer 1

∫xe^(x^2)dx

=(1/2)∫e^(x^2)dx^2

=(1/2)e^(x^2)

∫x^3e^(x^2)dx

=(1/2)∫x^2e^(x^2)dx^2

=(1/2)∫x^2de^(x^2)

=(1/2)x^2e^(x^2)-(1/2)∫e^(x^2)dx^2

=(1/2)x^2e^(x^2)-(1/2)e^(x^2)

所以原式=∫xe^(x^2)dx-∫x^3e^(x^2)dx

=(1/2)e^(x^2)-(1/2)x^2e^(x^2)+(1/2)e^(x^2)

=e^(x^2)-(1/2)x^2e^(x^2)（0到1）

=(e-e/2)-(1-0)

=e/2-1