/sci/ - Science & Math » Thread #14362561

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Why doesn't this work in reverse?

Anonymous Mon 04 Apr 02:21:42 2022 No.14362561 View Reply Original Report

Quoted By: >>14362720 >>14363480 >>14363505 >>14363579

Anonymous

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Anonymous Mon 04 Apr 2022 02:48:27 No.14362650 Report

Quoted By: >>14363523 >>14363579 >>14363592

I had to expand the binomial to get the full polynomial to integrate.

Anonymous

Anonymous Mon 04 Apr 2022 03:00:42 No.14362672 Report

Quoted By: >>14362676

Why does it matter though? What are you doing?

Anonymous

Anonymous Mon 04 Apr 2022 03:02:05 No.14362676 Report

Quoted By:

>>14362672
I'm just trying to figure out why it doesn't logically follow that the first equation is true if the second equation is true.

Anonymous

Anonymous Mon 04 Apr 2022 03:28:25 No.14362720 Report

Quoted By: >>14362728

>>14362561
Because the chain rule only works for derivatives. Integrals and derivatives aren’t symmetric operations

Anonymous

Anonymous Mon 04 Apr 2022 03:33:26 No.14362728 Report

Quoted By: >>14362855

>>14362720
Isn't u-substitution the inverse of the chain rule?

Anonymous

Anonymous Mon 04 Apr 2022 04:24:49 No.14362825 Report

Quoted By: >>14362832

Why should it work in the first place? You differentiated a different function and expect it to randomly be equal to the integral of another one?

Anonymous

Anonymous Mon 04 Apr 2022 04:27:56 No.14362832 Report

Quoted By:

>>14362825
This. I don’t get it.

Anonymous

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Anonymous Mon 04 Apr 2022 04:31:56 No.14362837 Report

Quoted By:

this person is stoned

Anonymous

Anonymous Mon 04 Apr 2022 04:36:13 No.14362855 Report

Quoted By:

>>14362728
yes, but sometimes u substitution doesn't work, that's why other rules for integration exist, but chain rule always work to get the derivative. in fact you could just solve each integration problem you get in HS/Uni by just brute forcing an antiderivative and seeing if its derivative equals your given equation

Anonymous

Anonymous Mon 04 Apr 2022 11:43:44 No.14363480 Report

Quoted By: >>14363484

>>14362561
Because you can take out a constant factor, but can't simply take out a variable. E.g.
$\dfrac{d}{dx}x^4=4x^3$
produces a factor of 4, which then allows you to take it when going in reverse:
$\int(x^3)\;dx=\frac{1}{4}x^4+c$
However, with a variable, those derivation and integration laws simply don't work:
$\dfrac{d}{dx}(\lambda f(x))=\lambda\dfrac{d}{dx}f(x)$
While:
$\dfrac{d}{dx}(x f(x))\neq\frac{1}{x}\dfrac{d}{dx}f(x)$
(analogical for integration rules)
So that's why you must find different methods to solve the integral.

Anonymous

Anonymous Mon 04 Apr 2022 11:45:37 No.14363484 Report

Quoted By:

>>14363480
Made a slight mistake in the final equation. This is of course what I meant to write:
$\dfrac{d}{dx}(x f(x))\neq x\dfrac{d}{dx}f(x)$

Anonymous

Anonymous Mon 04 Apr 2022 12:03:44 No.14363505 Report

Quoted By:

>>14362561
What are all these extra 2s floating around? Fix your handwriting

Anonymous

Anonymous Mon 04 Apr 2022 12:12:22 No.14363523 Report

Quoted By:

>>14362650
>I had to expand the binomial to get the full polynomial to integrate.
You didn't have to. You were just too stupid to solve it easily and made life harder for yourself.
$\int(x^4+5x)^2\;dx=\int x^2(x^3+5)^2\;dx=\frac{(x^3+5)^3}{9}+c$

Anonymous

Anonymous Mon 04 Apr 2022 12:40:48 No.14363579 Report

Quoted By:

>>14362650
>>14362561
Bc you can't do algebra

Anonymous

Anonymous Mon 04 Apr 2022 12:48:16 No.14363592 Report

Quoted By: >>14365619

>>14362650
The LCM of 9, 6 and 3 isn't 9 you fucking retard.

Anonymous

Anonymous Tue 05 Apr 2022 03:51:54 No.14365619 Report

Quoted By:

>>14363592
I wasn't looking for the LCM. It looks nicest when I use 9, as the leading term in the numerator doesn't have a coefficient before it.

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