## UnkleRhaukus 3 years ago question about notation

1. UnkleRhaukus

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

2. UnkleRhaukus

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

3. ash2326

As we know that integration is independent of variable. You could use any variable of your choice. I'd go with the first one :)

4. UnkleRhaukus

a?

5. ash2326

yes

6. UnkleRhaukus

$\int\limits_p^\infty F(a)\cdot\text da$

7. mukushla

$\int\limits_p^\infty F(x)\cdot\text dx$

8. waterineyes

$\int\limits\limits_p^\infty F(mukushla)\cdot\text d(mukushla)$

9. UnkleRhaukus

10. UnkleRhaukus

the notation is still troubling me

11. waterineyes

Where???

12. UnkleRhaukus

well i kinda had to change the question so i could answer it

13. UnkleRhaukus

\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*}

14. UnkleRhaukus

do you understand why i am getting confused/

15. waterineyes

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??

16. UnkleRhaukus

well the question in the book makes no sense

17. UnkleRhaukus

@sasogeek

18. sasogeek

certainly this is confusing, but why not use u-substitution for F(p) ?

19. UnkleRhaukus

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*}

20. sasogeek

$$let \ u=F(p)$$ $$\large \frac{du}{dp}=F'(p)$$ $$\large dp =\frac{du}{F'(p)}$$ $\int\limits_{p}^{\infty}u \ \frac{du}{F'(p)}$

21. UnkleRhaukus

im not sure why you'd do that?

22. sasogeek

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?

23. UnkleRhaukus

differential equations; laplace transforms

24. sasogeek

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 :)

25. UnkleRhaukus

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

26. UnkleRhaukus

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

27. sasogeek

you could ask satellite73 or amistre when they get online....

28. UnkleRhaukus

@amistre64

29. amistre64

yeah, i looked at this earlier and dont have anything pertinent to add to it :/ srry