Integral: Hint

  • Thread starter rocomath
  • Start date
  • #1
1,752
1
Integral: Hint plz

[tex]\int\frac{x\tan^{-1}x}{(1+x^2)^3}dx[/tex]

[tex]\int\frac{\tan^{ - 1}x}{(1 + x^2)}\frac{x}{(1 + x^2)^2}dx\left\{\begin{array}{cc}x = \tan{r} \leftrightarrow r = \tan^{ - 1}x \\
dr = \frac {dx}{1 + x^2}\end{array}\right\}[/tex]

Let's not even get into what I did next and how much work I put into this Integral. This is definitely not the right way to go, hmm ...

Ok I think I see it now ... omg.
 
Last edited:

Answers and Replies

  • #2
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
134
Well, therefore, you have:
[tex]\int\frac{x\tan^{-1}(x)}{(1+x^{2})^{2}}\frac{dx}{1+x^{2}}=\int\frac{\tan(r)r}{(1+\tan^{2}(r))^{2}}dr=\int\cos^{4}(r)\tan(r)rdr=\int{r}\cos^{3}(r)\sin(r)dr=\frac{1}{4}\int\cos^{4}(r)dr-\frac{r}{4}\cos^{4}(r)[/tex]

Agreed?
 
  • #3
1,752
1
On my previous try, after the 2nd to last step of yours, I did it differently; I would like to try your way, but I'm not sure what you did.

I did ... [tex]\int r\sin{r}(1-\sin^{2}r)\cos{r}dr[/tex]

Nvm, I see what you did. UHHH!!! I can't believe I spent more than an hour on this problem. Thanks arildno :-]
 
Last edited:
  • #4
154
0
Well, therefore, you have:
[tex]\int\frac{x\tan^{-1}(x)}{(1+x^{2})^{2}}\frac{dx}{1+x^{2}}=\int\frac{\tan(r)r}{(1+\tan^{2}(r))^{2}}dr=\int\cos^{4}(r)\tan(r)rdr=\int{r}\cos^{3}(r)\sin(r)dr=\frac{1}{4}\int\cos^{4}(r)dr-\frac{r}{4}\cos^{4}(r)[/tex]

Agreed?

Hi arildno,

I followed all but how you went from the second last step to the last step, as there is an 'r' in the integrand. Are you using some sort of gamma function there. That's the closest I've seen to int( sin^m(x) x cos^n(x) dx)

Thanks.
 
  • #5
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
134
I'm just using the product rule:
[tex]u(r)=r, v'(r)=\cos^{3}(r)\sin(r)\to{v}(x)=-\frac{1}{4}\cos^{4}(r)[/tex]
 
  • #6
154
0
I'm just using the product rule:
[tex]u(r)=r, v'(r)=\cos^{3}(r)\sin(r)\to{v}(x)=-\frac{1}{4}\cos^{4}(r)[/tex]

You're a freakin genius. I totally didn't see that. I was trying to solve it using a gamma function. I think there should be something related to a gamma function too as you still have to solve the int(cos^4(r) dr) which is part of the question.

Thanks for the help.
 

Related Threads on Integral: Hint

  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
21
Views
1K
  • Last Post
Replies
24
Views
3K
  • Last Post
Replies
5
Views
2K
Replies
2
Views
1K
Replies
14
Views
2K
  • Last Post
Replies
6
Views
2K
Replies
4
Views
1K
  • Last Post
Replies
1
Views
1K
Top