Happy new year everyone!
I have a question that I struggle related to integration by parts.
Suppose I'm going to solve the following using integration by parts. $$\int\limits^{2}_{0}pydy$$ Where $p$ is a function of $y$.
I know the formula for integration by parts for definite integrals:
$$\int\limits_{a}^{b}udv=uv\Big|_{a}^{b} - \int\limits_{a}^{b}vdu$$
Thus I'm considering: $$u=y \text{ and } dv=pdy$$
This gives me:
$$du=dy \text{ and } v=\int pdy$$
Now by substituting into the original formula,
I have:
$$\begin{align}
\int\limits^{2}_{0}pydy &= \left(y\int pdy\right)\Big|_{0}^{2} - \int\limits_{0}^{2}(\int pdy)dy
\end{align}$$
I also have the function $Q(y)$ where, $$Q(y) = \int\limits_{0}^{y}p(t)dt$$
Now this is the place I find it confusing to substitute the limits of the integral.
And my goal is to write the integration by parts equation in terms of this function $Q(y)$ because $Q(y)$ is actually a two-dimensional flow rate where we have $Q(0)=0$, $Q(2)=k$ a known value.
Appreciate your help