曲线积分中的全微分

Question

为什么2x的积分结果x^2没了

Answer 1

全微分方程
(3yx^2+8xy^2)dx + (x^3 + 8yx^2 + 12ye^y)dy = 0
通解是 u(x,y) = ∫<0, x>(3yx^2+8xy^2)dx + ∫<0, y> (0^3 + 8y 0^2 + 12ye^y)dy
= yx^3 + 4x^2y^2 + 12∫<0, y> yde^y
= yx^3 + 4x^2y^2 + 12ye^y -12∫<0, y> e^ydy
= yx^3 + 4x^2y^2 + 12ye^y - 12e^y + 12 = C