# 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
9 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $\boxed{123}$

Sort by:

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.

- 9 months ago

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

- 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.

- 9 months ago

@Mark Hennings Thank you so much ,

- 9 months 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."

- 9 months 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.

- 9 months 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.

- 9 months 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 .

- 9 months 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.

- 8 months, 4 weeks ago

This is the perfect solution.

- 9 months ago

@Pi Han Goh please can we have some alternative approach

- 9 months ago

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

- 9 months ago

- 9 months ago

- 9 months ago

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

- 9 months ago

@Krishna Karthik Because its value is that only

- 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

- 9 months ago

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

- 9 months ago

Yea mate. Thanks.

- 8 months, 4 weeks ago