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# STEP Tricks and Hacks Thread watch

1. (Original post by Zacken)
just a few suggestions, you might want to add in what's wrong with the AM-GM inequality "proof" that you showed and how to fix it
Will edit now.
(Original post by Zacken)
P.S: Nice touch adding the proof by contrapositive, I just want to add my thoughts on it for anybody who doesn't see why
I'm a big fan of the contrapositive
2. (Original post by joostan)
Will edit now.
Thanks!

I'm a big fan of the contrapositive
As am I - I've only recently started learning off the logic stuff and it's often much easier to do then by making one of them the contrapositive because it's rare to have both directions of the implication be easy.

On the other hand... there's many a time I've skipped a step and done and then instead of and realise that I've proven the same thing twice - massive facepalms.
3. (Original post by Zacken)
One recurring theme is to exploit the symmetry of trigonometric functions during integrating. Using this in conjunction with:

is particularly useful.

Here's an example: . How is this relevant here? Call our integral , then using the substitution we have:

Extension to this, using the same idea, roughly - attempt
Am I on the right track?

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4. (Original post by Insight314)
Am I on the right track?

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You messed up your negative signs, you should get with the limits from to on each integral but I can't be arsed typesetting.

Then you should add both integrals to get with the usual limits and then think about .
5. (Original post by Zacken)
You messed up your negative signs, you should get with the limits from to on each integral but I can't be arsed typesetting.

Then you should add both integrals to get with the usual limits and then think about .
But doesn't the substitution change the limits from 0 to pi/2, to pi/2 to 0 so then I switch it by adding a negative sign.

I actually did it that way the first time I tried the integral but then I noticed that the substitution changes the limits.

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6. (Original post by Insight314)
But doesn't the substitution change the limits from 0 to pi/2, to pi/2 to 0 so then I switch it by adding a negative sign.

I actually did it that way the first time I tried the integral but then I noticed that the substitution changes the limits.

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Yes - but you forgot that adds in a factor of since . This is why I would caution using the notation instead of at this stage. You can use the fancy notation when you're more comfortable with everything.
7. (Original post by Zacken)
Yes - but you forgot that adds in a factor of since .
Oh **** yes. Also, I remember that then you get an integral with ln 2 or something like that as integrand which is easy to calculate but the other integral with an integrand and then you do the same addition of 1/2 integral of cos + 1/2 integral of sin but with this time and then you continue ad infinitum. How does this simplify it? I can't find the workings out on this integral I did yesterday, so I might be wrong.
8. (Original post by Insight314)
Oh **** yes. Also, I remember that then you get an integral with ln 2 or something like that as integrand which is easy to calculate but the other integral with an integrand and then you do the same addition of 1/2 integral of cos + 1/2 integral of sin but with this time and then you continue ad infinitum. How does this simplify it? I can't find the workings out on this integral I did yesterday, so I might be wrong.

But you want an in there somewhere, so the sub () gets you that and some nice thinking about the symmetry of sine gets you the home run.
9. (Original post by Zacken)

But you want an in there somewhere, so the sub () gets you that and some nice thinking about the symmetry of sine gets you the home run.
I got
10. (Original post by Insight314)
I got
Haha, you're the third person to give me that incorrect answer since I've posted that problem.
11. (Original post by Zacken)
Haha, you're the third person to give me that incorrect answer since I've posted that problem.

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12. The limits change after the sub .
13. (Original post by Zacken)
We know that is a true statement literally every single time unless we have the case that is false and is true, we can rewrite this as: is a true statement unless .
Look at some truth tables. https://en.wikipedia.org/wiki/Truth_table

The conditional connective, is false, if, and only if the premise is true but the conclusion is false.
14. (Original post by Zacken)
The limits change after the sub .
And I did change them? Then I split the integral into limits consisting of each.
15. (Original post by zetamcfc)
Look at some truth tables. https://en.wikipedia.org/wiki/Truth_table

The conditional connective, is false, if, and only if the premise is true but the conclusion is false.
That's what I meant, I'll edit that. Thanks.
16. (Original post by Insight314)
And I did change them? Then I split the integral into limits consisting of each.
Oh, yes.

You seem to be saying that .

You want to say that
17. (Original post by Zacken)
Oh, yes.

You seem to be saying that .

You want to say that
which is where this 1/2 comes from.
18. (Original post by Insight314)
which is where this 1/2 comes from.
But you multiplied the entire thing by the line before.

We have

You decided to multiply everything by to get: .

But we know that the last term is so
19. (Original post by Zacken)
But you multiplied the entire thing by the line before.

We have

You decided to multiply everything by to get: .

But we know that the last term is so
Oh yes, I saw my mistake. I think that was more of a silly mistake to be quite honest.

This was a really fun question, thanks a lot.
20. (Original post by Insight314)
Oh yes, I saw my mistake. I think that was more of a silly mistake to be quite honest.

This was a really fun question, thanks a lot.
It was.

No problem! I'll try and think up some more when I've got some free time on my hands. :-)

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