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Do we need to memorise this identity? (3 Viewers)

SammyT123

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Thanks paradoxica

I personally prefer the trig substitution

But jeez that method is also very quick :D
 

SammyT123

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Are the t-formula results ever 'unavoidable'

(I know that theoretically they are obviously unavoidable, but sitting in the middle of a 4U hsc exam is it still feasible to avoid them?)
 

leehuan

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Are the t-formula results ever 'unavoidable'

(I know that theoretically they are obviously unavoidable, but sitting in the middle of a 4U hsc exam is it still feasible to avoid them?)
Something like this is just dodge

 

Paradoxica

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Also one of the things to keep in mind for the tangent half angle substitution is that you have to take care when integrating across a singularity....

Because that can give different results if you ignore the singularity versus you splitting and taking limits.
 

Paradoxica

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Something like this is just dodge

Also

1. Yuck. The numbers aren't even nice. Would have been lovely to have 3sinx + 4cosx but nooooooooo you didn't give me a correctly ordered pythagorean triple

2. t-formula is messy answer anyway, since it involves a fair bit of surds

3. *flies away*
 

SammyT123

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Also

1. Yuck. The numbers aren't even nice. Would have been lovely to have 3sinx + 4cosx but nooooooooo you didn't give me a correctly ordered pythagorean triple

2. t-formula is messy answer anyway, since it involves a fair bit of surds

3. *flies away*
Went in balls deep
Didn't seem too bad unless I screwed up completely LOL
 

Paradoxica

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Went in balls deep
Didn't seem too bad unless I screwed up completely LOL
I had √2, √3 appear in the answer.

If I see more than one non-square number under a square root, I will immediately back out and look to see if there is a better path to the answer.

Otherwise, proceed through the flames of hell.
 

SammyT123

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I had √2, √3 appear in the answer.

If I see more than one non-square number under a square root, I will immediately back out and look to see if there is a better path to the answer.

Otherwise, proceed through the flames of hell.


What did I do wrong this time
Rip
 

Drongoski

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I recall doing a similar integral a few days ago, from Bob Aus's book.

Using the usual t-formulae:


 
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Paradoxica

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I recall doing a similar integral a few days ago, from Bob Aus's book.

Using the usual t-formulae:


The Auxiliary Transformation and the Tangent Half Angle Substitution will be guaranteed to be nice for (asinx + bcosx + c)-1 IFF (a,b,c) is a Pythagorean Triple.
 

Drongoski

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The Auxiliary Transformation and the Tangent Half Angle Substitution will be guaranteed to be nice for (asinx + bcosx + c)-1 IFF (a,b,c) is a Pythagorean Triple.
That's interesting. I have not investigated, but I noticed that in both integrals, {3,4,5} and {5,12,13} are Pythagorean Triples.
 

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