Integration

[proper plover]

So I've solved it, but how exactly would you guys approach it? (Please share thought processes aswell)

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[proper plover]

Question 26

[ionic path]

@proper plover I guess, being a multi - you could just differentiate each option.

[proper plover]

thats what i have been doing this whole time

but what if an integral comes in the form of a numerical type question

ill be doomed

[ionic path]

Fair enough. Have you tried substitution?

perhaps u = sqrt(x+1) ?

[proper plover]

Please read my entire thread, ive solved the question

But i want to know what exactly people think when approaching an integration question

And why.

[meager meadow]

If you're integrating a function like this (where R(u, v) is a rational function), then the standard approach is to do ax + b = u^n, as that rationalizes the expression.

[proper plover]

Is this called euclid substitution or smth?

[meager meadow]

If you're talking about Euler substitution, then no.

This one doesn't have a name.

[proper plover]

Ouhh

Okayy

Ill try questions wit this format

[meager meadow]

Oh, by the way, this isn't the most general form of this substitution.
Though, probably try doing some simpler questions first.

[proper plover]

Ive done quite some questions

I think i already did simpler ones

[meager meadow]

Ah, alright. One sec, then.

[proper plover]

Oka

[meager meadow]

So, the general case is this. Here R is a rational function.
Notice that the expressions under the roots are the same. Then, if n = lcm(n1, n2, ... , nk), take u^n = (ax + b)/(Ax + B). This also rationalizes the whole thing.

[proper plover]

YESSSSSSS

We studied the

Lcm thing

[meager meadow]

Oh, yeah, actually. Meant LCM.

Corrected.

[proper plover]

So that all the roots become integral powers

[meager meadow]

So, for example, if you want to integrate 1/(x + √((x + 1)/(x - 1)) - 5∛((x + 1)/(x - 1))), we have n1 = 2, n2 = 3. So, n = lcm(2, 3) = 6 and the substitution is u^6 = (x + 1)/(x - 1).

Yeah.

[proper plover]

I had studied this formaaaaaaat

But i couldn't see it in the question

Thankyou ❤️

[meager meadow]

You're welcome!

[proper plover]

i have a doubt

are all the radicals supposed to have the same thing inside?

[meager meadow]

Yes.

[proper plover]

oof so i'll have to think of something else for this question 💔

[meager meadow]

Which one?

[proper plover]

im doing one rn, havent sent it here

expect me to ping you for help- 😭

anyways thankyouu, ill close this thread now

[meager meadow]

Oh, alright. You're welcome!

[proper plover]

+close

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