#### blyatman

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Yes, please see one of the subsequent posts.These are all beyond the scope of the syllabus.

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Yes, please see one of the subsequent posts.These are all beyond the scope of the syllabus.

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

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Have anyone figured out?This one is challenging if you can't think of a useful substitution.

The trick is:

Clearly, the useful substitution is

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

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

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intuition said it was φ-like but i didn't want to stay up that late to figure it out;Clearly, the useful substitution is

a good one nonetheless

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).

These are Mr Blyatman's question. Ill work them up later today

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How are you supposed to know phi is a good sub here? I mean looking at the original question aloneHave anyone figured out?

The trick is:

Clearly, the useful substitution is

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Inspection.How are you supposed to know phi is a good sub here? I mean looking at the original question alone

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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.How are you supposed to know phi is a good sub here? I mean looking at the original question alone

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Has anyone attempted this?This one is tedious.

To get the final answer, you have to express the annoying tangent in surd form.

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Feel free to share your attempt.

That... doesn't necessarily have to be true, but my intuition agrees with that.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.

Formally, however, I do not.

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MIT ???

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No attempt for 2 weeksFeel free to share your attempt.

Is it too tedious? I think it's a good practice for algebraic skills.

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Math is not really my area but I tried nonethelessFeel free to share your attempt.

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Math is not really my area but I tried nonetheless

"Math is not my area",,,

Bro that question legit took an hour to solve

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Unfortunately, there's a mistake in this piece of integral.

Fortunately, it doesn't affect the final answer.

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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

Unfortunately, there's a mistake in this piece of integral.

Fortunately, it doesn't affect the final answer.

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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.