In this case..first we observe that the nested radicals are of the form:21+2a which suggests that a can be considered as cosθ for some θ..Thus the nested radical part of the integral reduces to 21+2121+2121+...+2121+2cosθ=cos(2nθ) where cosθ=x2+(lncosx)2(lncosx)2⇒sec2θ=(lncosx)2x2+(lncosx)2=1+(lncosx)2x2=1+tan2θ⇒tanθ=±lncosxx We choose θ to be such that tanθ=−lncosxx. Now, let r∈(0,1). We use the two following results:∫0+∞yr−1cos(ay)e−bydy=Γ(r)(a2+b2)2rcos{rtan−1(ba)} and ∫02πcos(yx)cosyxdx=2y+1π which are trivial.
Then we can write ∫02π{x2+(lnx)2}2rcos{rtan−1(−lncosxx)}dx=Γ(r)1∫02π{∫0+∞yr−1cos(xy)eylncosxdy}dx=Γ(r)1∫02π∫0+∞yr−1cos(xy)eylncosxdydx=Γ(r)1∫0+∞yr−1{∫02πcos(xy)eylncosxdx}dy=Γ(r)1∫0+∞yr−1{∫02πcos(xy)cosyxdx}dy=Γ(r)1∫0+∞yr−12y+1πdy=2Γ(r)π∫0+∞yr−12−ydy=2Γ(r)π∫0+∞yr−1e−yln2dy=2Γ(r)π.(ln2)rΓ(r)=2(ln2)rπ (clearly in steps 2 and 3 we have used Fubini's theorem and in step 2 I have used gamma function since r>0.)
We can apply analytic continuation to the above result to extend the domain of r from (0,1) to (−1,1) since the original domain (0,1) is open in R and the integrand function is analytic in r. Now we simply put r=−2n1 to get that Rn+=π2.2(ln2)−2n1π=(ln2)2−n.
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.
Markdown
Appears as
*italics* or _italics_
italics
**bold** or __bold__
bold
- bulleted - list
bulleted
list
1. numbered 2. list
numbered
list
Note: you must add a full line of space before and after lists for them to show up correctly
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:
*italics*
or_italics_
**bold**
or__bold__
paragraph 1
paragraph 2
[example link](https://brilliant.org)
> This is a quote
\(
...\)
or\[
...\]
to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
@Surya Prakash