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.

Thanks in advance.

#Calculus

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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

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.

Mark Hennings - 9 months ago

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

Talulah Riley - 9 months 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.

Mark Hennings - 9 months ago

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

Talulah Riley - 9 months ago

@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 - 9 months ago

@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 - 9 months ago

@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 - 9 months ago

@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 - 9 months ago

@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 - 9 months ago

This is the perfect solution.

Pi Han Goh - 9 months ago

@Pi Han Goh please can we have some alternative approach

Talulah Riley - 9 months ago

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

Talulah Riley - 9 months ago

I've explained how I've solved it here

Pi Han Goh - 9 months ago

@Steven Chase @Karan Chatrath @Mark Hennings @Krishna Karthik

Talulah Riley - 9 months ago

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

@Lil Doug

Krishna Karthik - 9 months ago

@Krishna Karthik Because its value is that only

Talulah Riley - 9 months 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

Krishna Karthik - 9 months ago

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

Talulah Riley - 9 months ago

@Talulah Riley – Yea mate. Thanks.

Krishna Karthik - 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$