integral of integral of e^{x/y}

解

\int \int e^{\frac{x}{y}} d x d y

C y + \frac{1}{2} e^{\frac{x}{y}} y^{2} + \frac{x e ^{\frac{x}{y}} y}{2} - x \frac{x}{2} Ei (\frac{x}{y}) + C

解答ステップ

\int \int e^{\frac{x}{y}} d x d y

\int e^{\frac{x}{y}} d x = y e^{\frac{x}{y}} + C

= \int (y e^{\frac{x}{y}} + C) d y

\int (y e^{\frac{x}{y}} + C) d y = C y + \frac{1}{2} e^{\frac{x}{y}} y^{2} + \frac{x e ^{\frac{x}{y}} y}{2} - x \frac{x}{2} Ei (\frac{x}{y}) + C

= C y + \frac{1}{2} e^{\frac{x}{y}} y^{2} + \frac{x e ^{\frac{x}{y}} y}{2} - x \frac{x}{2} Ei (\frac{x}{y}) + C