Tricky Integration

Here is a problemπ2π(x2+2)cosxx3dx\large \int_{\pi}^{2\pi} \frac{(x^2+2) \cos x}{x^{3}} \, dx
Here is the solution


Guys if anyone has an alternative approach please share it in comments below.

Thanks in advance.

Note by Talulah Riley
1 month ago

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Or, more simply, ddx(sinxxcosxx2)  =  x2+2x3cosx \frac{d}{dx}\left(\frac{\sin x}{x} - \frac{\cos x}{x^2}\right) \; = \; \frac{x^2 + 2}{x^3}\cos x so the integrand is exact.

Mark Hennings - 1 month ago

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@Mark Hennings But how do you know that? It is difficult for me to predict

Talulah Riley - 1 month ago

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Elegant proofs are often the result of earlier work. Once the integration by parts cancelled out all the integrals, all it takes is to go through your result and write a shiny, single-line, proof. On the other hand, for this integral to have a nice simple answer (not expressed in terms of the cosine and sine integrals and their relatives, it was highly likely that an exact integral was hiding here, so I looked for it.

Mark Hennings - 1 month ago

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@Mark Hennings @Mark Hennings Thank you so much ,
Can you please help me in this problem,
Thanks in advance.

Talulah Riley - 1 month ago

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@Talulah Riley Hey buddy, it's actually better to show your attempt as well. Otherwise, your comment might come off as "Hey, do my homework for me because I'm lazy."

Pi Han Goh - 1 month ago

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@Pi Han Goh @Pi Han Goh It is not my homework.
I am solving , because I have my own interest.
I will post a note showing the attempt of this problem within some hours.

Talulah Riley - 1 month ago

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@Pi Han Goh I can verify that 9/10 times when he asks a doubt about a question, he tries the question out himself.

And, he's not assigned this as homework, so I wouldn't know why else he'd be doing this, unless he wants to personally benefit and learn how to solve these kinds of questions.

This stuff is for his JEE preparation.

Krishna Karthik - 1 month ago

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@Krishna Karthik @Krishna Karthik For not Jee bro. It's my passion.
And it is good that Jee has included physics in their syllabus , so it is basically a benefit for me .

Talulah Riley - 1 month ago

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@Talulah Riley See, this is why I admire you. You take the effort to do all this physics, and better your ability to apply your knowledge. Good stuff mate. Don't listen to Pi Han Goh.

Krishna Karthik - 1 month ago

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This is the perfect solution.

Pi Han Goh - 1 month ago

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@Pi Han Goh please can we have some alternative approach

Talulah Riley - 1 month ago

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@Pi Han Goh i think you said A,B,C,DA, B, C, D, you were talking about some different approach?

Talulah Riley - 1 month ago

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Ummm why is my integral calculator coming up with a different value, 0.12-0.12?

@Lil Doug

Krishna Karthik - 1 month ago

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@Krishna Karthik Because its value is that only

Talulah Riley - 1 month ago

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WTFFFFFF

I actually solved the problem then! I tried simplifying and then it came up with 0.03!!!! WTF is actually going onnnnn... I used integration by parts only, btw.

And I've been thinking like an idiot that all this time I didn't solve it right... lmao

Krishna Karthik - 1 month ago

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@Krishna Karthik @Krishna Karthik nice, so this time you solved it. Well done!

Talulah Riley - 1 month ago

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@Talulah Riley Yea mate. Thanks.

Krishna Karthik - 1 month ago

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