correct integration notation (1 Viewer)

Aeonium · Jan 2, 2024

let's say i want to get the original function given the equation of the 2nd derivative:
$\frac{d^2y}{dx^2}\,3x^2-2x+1$
how would you notate it properly? this is what i have rn; idk if it's correct
$y=\int\left(\int 3x^2-2x+1\,dx\right)\,dx+ C\\$

Average Boreduser · Jan 2, 2024

Aeonium said:
let's say i want to get the original function given the equation of the 2nd derivative:
$\frac{d^2y}{dx^2}\,3x^2-2x+1$
how would you notate it properly? this is what i have rn; idk if it's correct
$y=\int\left(\int 3x^2-2x+1\,dx\right)\,dx+ C\\$

Just say dydx=S…
Then the next line with the parent function

liamkk112 · Jan 2, 2024

Aeonium said:
let's say i want to get the original function given the equation of the 2nd derivative:
$\frac{d^2y}{dx^2}\,3x^2-2x+1$
how would you notate it properly? this is what i have rn; idk if it's correct
$y=\int\left(\int 3x^2-2x+1\,dx\right)\,dx+ C\\$

better to do this:
$\frac{dy}{dx}\ = \int(3x^2-2x+1)\,dx = x^3-x^2+x+C$
then,
$y = \int(x^3-x^3+x+C)\,dx = \frac{1}{4}\ x^4-\frac{1}{3}\ x^3+\frac{1}{2}\ x^2+Cx+D$

this shows working out better and much more clearly shows what is happening. also, given that you wrote a +C already, its not quite clear about the second constant, as the equation is second order there should be two unknown constants, and your integration would not include the term of Cx present in the solution.

Luukas.2 · Jan 3, 2024

I agree with @liamkk112's advice. Double integrals, with one integral sign after another, have specific meaning that is beyond HSC Maths, so I advise avoiding notation like you have written and just solve the DE by two separate integration steps.

ZakaryJayNicholls · Jan 16, 2024

Aeonium said:
let's say i want to get the original function given the equation of the 2nd derivative:
$\frac{d^2y}{dx^2}\,3x^2-2x+1$
how would you notate it properly? this is what i have rn; idk if it's correct
$y=\int\left(\int 3x^2-2x+1\,dx\right)\,dx+ C\\$

$y=\int\left(\int 3x^2-2x+1\,dx\right)\,dx\\$
The constants of integration will appear after the integration is completed, not prior. Apart from this, it is actually perfectly fine notation. (This kind of integral nesting is not an uncommon style of notation in applied mathematics and physics, in some cases where nth integrals are not used).

It was suggested that denoting dy/dx=S would be better, this is a very good suggestion.

wizzkids · Jan 17, 2024

I would have written it this way, it seems a bit clearer to me, and yes the constant for the definite integral should not appear at this stage:
$y=\int\int(3x^2-2x+1)\,dx\, dx$

correct integration notation (1 Viewer)

Aeonium

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Average Boreduser

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liamkk112

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Luukas.2

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ZakaryJayNicholls

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wizzkids

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