HSC 2018-2019 MX2 Integration Marathon (1 Viewer)

stupid_girl

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I have a feeling this is not the fastest method...









(the proof is left as an exercise to the reader.)



From here, I will use the result , which is not difficult to verify via the appropriate double angle identity.











Note that the absolute value is important when evaluating the lower bound.





My method is similar to yours. DI table may make the integration by parts neater.






 

stupid_girl

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This one is relatively routine.
This integral itself should be quite routine. Getting the final answer in terms of pi is slightly tricky.







To get the final answer, you need to show that:
 
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blyatman

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These are Mr Blyatman's question. Ill work them up later today
As it turns out, the indefinite counterparts of some of these integrals have no closed-form solutions (I tried googling a solution to no.9), so some of these definite integrals must be solved (assuming they can be solved conventionally) without directly integrating into the anti-derivative, similar to how the integral of exp(x)cos(x) is done by using integration by parts twice (so that we have 2*integral = something).
 

stupid_girl

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How are you supposed to know phi is a good sub here? I mean looking at the original question alone
After dividing top and bottom by 27^x, the integral is in the form 1/(p^x+q^x+r^x). If this integral has an elementary form, then there must be a relationship between p,q and r. After some trial and error, you would get that relationship.
 

Paradoxica

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If this integral has an elementary form, then there must be a relationship between p,q and r. After some trial and error, you would get that relationship.
That... doesn't necessarily have to be true, but my intuition agrees with that.

Formally, however, I do not.
 

vernburn

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Just responding to an earlier integral based on the infinite expansion for e. I just discovered this thread and i love it!
 

Drdusk

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Good try!
Unfortunately, there's a mistake in this piece of integral.😣
Fortunately, it doesn't affect the final answer.:p
yeah I kinda realized that while going to sleep but I have no idea how to integrate sinx/x. so I kinda just went meh
 

stupid_girl

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It should be well known enough that

and

(can be derived by integration by parts twice)

This is how you may make the life easier.




Overall speaking, the integral requires log rule, trig rule, 2 substitutions, 3 times integration by parts and algebraic manipulations...nice practice for MX2.😈
 

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