119. Double Integrals

119.0.1. 1. Double Indefinite Integral

First Integration:

Integrate with respect to $𝑥$ (treat $𝑦$ as constant):

• $𝐹\left(𝑥,𝑦\right)$: antiderivative of $𝑓$ w.r.t. $𝑥$
• ${𝐶}_1\left(𝑦\right)$: Constant of integration (can depend on $𝑦$)

Second Integration:

Integrate with respect to $𝑦$ (treat $𝑥$ as constant):

• $𝐺\left(𝑥,𝑦\right)$: antiderivative of $𝐹\left(𝑥,𝑦\right)$ with respect to $𝑦$

Final Output:

A family of functions $𝐺\left(𝑥,𝑦\right)+{𝐶}_2$, representing all functions whose mixed partial derivative $\frac{{𝜕}^2}{𝜕𝑦𝜕𝑥}$ is $𝑓\left(𝑥,𝑦\right)$