Here's the question you clicked on:
UnkleRhaukus
question about notation
in my book there is a question involving a term like this \[\int\limits_p^\infty F(p)\cdot\text dp \], but this is a confusing choice of notation because the variable of integration is one one of limits of integration
as far as i know the convention is to change to \[\int\limits_p^\infty F(p')\cdot\text dp'\] but i dont like this because it looks like the prime of p
As we know that integration is independent of variable. You could use any variable of your choice. I'd go with the first one :)
\[\int\limits_p^\infty F(a)\cdot\text da\]
\[\int\limits_p^\infty F(x)\cdot\text dx\]
\[\int\limits\limits_p^\infty F(mukushla)\cdot\text d(mukushla)\]
the notation is still troubling me
well i kinda had to change the question so i could answer it
\[\begin{align*} \int\limits_p^\infty F(q)\cdot\text dq &=\int\limits_p^\infty\int\limits_0^\infty f(x)e^{-qx}\cdot\text dx\cdot\text dq \\&=\int\limits_0^\infty f(x)\int\limits_p^\infty e^{-qx}\cdot\text dq\cdot\text dx \\&=\int\limits_0^\infty f(x)\left.\frac{e^{-qx}}{-x}\right|_p^\infty \cdot\text dx \\&=\int\limits_0^\infty \frac{f(x)}xe^{-px}\cdot\text dx \\&=\mathcal L\left\{\frac {f(x) }x \right\} \end{align*}\]
do you understand why i am getting confused/
The things you are doing are going above my head, but I am not understanding why are you getting confused, where are you not feeling comfortable??
well the question in the book makes no sense
certainly this is confusing, but why not use u-substitution for F(p) ?
like this? \[\begin{align*} \int\limits_p^\infty F(p)\cdot\text dp\\ \text{let }u=p\\ \text du=\text dp\\ \\p=p\rightarrow u=u\\ p=\infty\rightarrow u=\infty\\ \\&=\int\limits_u^\infty F(u)\cdot\text du\\ \end{align*}\]
\(let \ u=F(p)\) \(\large \frac{du}{dp}=F'(p)\) \(\large dp =\frac{du}{F'(p)} \) \[\int\limits_{p}^{\infty}u \ \frac{du}{F'(p)}\]
im not sure why you'd do that?
well having the function variable being the same as one of the limits... i'm not quite sure how you'd go about solving it but thats the first thing that comes to my mind. besides, i'm not so good at calculus, but that's what i'd do. it may or may not be right, but it's worth a shot :/. I'm pretty certain this isn't calculus 1... right?
differential equations; laplace transforms
there we go, that's beyond me lol, but all things being equal, thats what i'd do. sorry i wasn't of much help lol :/ but i'll stick around to learn :)
but the problem im having is not the content of the question, i think i have answered that satisfactorily , my issus is the choice of notation used in the question
i dont think the question makes sense as written \[....=\int\limits_p^\infty F(p)\text dp\] it dosent make sense to have the variable as a limit of integration
you could ask satellite73 or amistre when they get online....
yeah, i looked at this earlier and dont have anything pertinent to add to it :/ srry