Integration problems, help!

Prove that,∫0∞1x3sin⁡3(x−1x)5 dx=0\int_{0}^{\infty}\dfrac{1}{x^3}\sin^3\left(x-\dfrac{1}{x} \right)^5\, dx=0 and∫−10x2+2xln⁡(x+1)dx=ln⁡3\int_{-1}^{0}\dfrac{x^2+2x}{\ln{(x+1)}} dx=\ln{3}.I have no idea how to solve these problems!

#Calculus #Help #Integrationdefinite

Note by Adarsh Kumar
5 years, 11 months ago

No vote yet
1 vote

  Easy Math Editor

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.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list
  • bulleted
  • list

1. numbered
2. 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 1

paragraph 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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai−1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 ∑i=13 \sum_{i=1}^3
\sin \theta sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Hello Adarsh. I'm really sorry but I won't be able to help now; I'm truly exhausted after coaching class since 10 in the morning - and the same schedule is in place for tomorrow. I'll answer as soon as I can for sure.

User 123 - 5 years, 11 months ago

Log in to reply

No problem bhaiya!I understand!

Adarsh Kumar - 5 years, 11 months ago

Hint about second integral, take x+1=t, and do it with the help of differentiation under integral.

Ronak Agarwal - 5 years, 11 months ago

Log in to reply

Actually i tried that and I got,∫01t2−1ln⁡tdt\int_{0}^{1}\dfrac{t^2-1}{\ln{t}}dt,how to proceed now?

Adarsh Kumar - 5 years, 11 months ago

@Adarsh Kumar Seems you've got some excellent answers. I would have done the second the same way as Rajdeep.

User 123 - 5 years, 11 months ago

Log in to reply

Sir the 1st one ? @Samarpit Swain solution is wrong.

Rajdeep Dhingra - 5 years, 11 months ago

Log in to reply

Come on Rajdeep! I'm not Sir!I'll have a go at it in half an hour.

User 123 - 5 years, 11 months ago

Log in to reply

@User 123 – Yes sir U are "sir" for me.

Rajdeep Dhingra - 5 years, 11 months ago

By the way really nice solution!

User 123 - 5 years, 11 months ago

Log in to reply

@User 123 – Thank You sir

Rajdeep Dhingra - 5 years, 11 months ago

@Adarsh Kumar @Rajdeep Dhingra The first Integral is really hard. Also, it is not equal to 0.0. The Integral is equivalent to −15∫0∞sin⁡3(u)u−3/5du≈−0.1535-\frac15 \int_0^\infty \sin^3(u) u^{-3/5} du \approx {-0.1535} Anyway, to be able to compute the original Integral, you need to know some conditions for which an Integral becomes Riemann Improper Integrable. I can try to proceed if you want, but this is really miles and miles above our level.

User 123 - 5 years, 11 months ago
×

Problem Loading...

Note Loading...

Set Loading...