求∫(1,0)x^2dx∫(1,x)e^-y^2dy

Question

求∫ (1,0)x^2dx∫(1,x) e^-y^2dy

Answer 1

∫(0到1) dx ∫(x到1) x²e^(-y²) dy
需要交换积分次序
x = 0 到 x = 1,y = 1 到 y = x
∫(0到1) dy ∫(0到y) x²e^(-y²) dx
= ∫(0到1) [e^(-y²) * x³/3]|(0到y) dy
= (1/3)∫(0到1) y³e^(-y²) dy
(u = y²,du = 2y dy)
= (1/3)∫(0到1) u*ye^(-u) * du/(2y)
= (1/6)∫(0到1) ue^(-u) du
= (-1/6)∫(0到1) u d[e^(-u)]
= (-1/6)[ue^(-u)]|(0到1) + (1/6)∫(0到1) e^(-u) du
= (-1/6)e^(-1) - (1/6)[e^(-u)]|(0到1)
= -1/(6e) - (1/6)[e^(-1) - 1]
= -1/(6e) - (1 - e)/(6e)
= (e - 2)/(6e)