# Tricky Integration

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

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

Note by Talulah Riley
4 months, 1 week ago

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

- 4 months, 1 week ago

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

- 4 months, 1 week ago

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.

- 4 months, 1 week ago

@Mark Hennings Thank you so much ,

- 4 months, 1 week ago

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."

- 4 months, 1 week ago

@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.

- 4 months, 1 week ago

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.

- 4 months, 1 week ago

@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 .

- 4 months, 1 week ago

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.

- 4 months, 1 week ago

This is the perfect solution.

- 4 months, 1 week ago

@Pi Han Goh please can we have some alternative approach

- 4 months, 1 week ago

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

- 4 months, 1 week ago

- 4 months, 1 week ago

- 4 months, 1 week ago

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

- 4 months, 1 week ago

@Krishna Karthik Because its value is that only

- 4 months, 1 week ago

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

- 4 months, 1 week ago

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

- 4 months, 1 week ago

Yea mate. Thanks.

- 4 months, 1 week ago