All Questions
4
questions
2
votes
1
answer
100
views
Find the limit and integral $\lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{x^4 + x^2 + 1} \, dx $
Find the limit and integral$$ \lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{x^4 + x^2 + 1} \, dx $$
My try
$$
\lim_{\epsilon \to 0} \int_{\epsilon}^{1} \frac{x \sqrt{x} \log(x)}{...
6
votes
1
answer
261
views
Find $\lim_{a\to \infty}\frac{1}{a}\int_0^{\infty}\frac{x^2+ax+1}{1+x^4}\cdot\arctan(\frac{1}{x})dx$
Find
$$
\lim_{a\to \infty}
\frac{1}{a}
\int_0^{\infty}\frac{x^2+ax+1}{1+x^4} \arctan\left(\frac{1}{x}\right)dx
$$
I tried to find
$$
\int_0^{\infty} \frac{x^2+ax+1}{1+x^4}\arctan\left(\frac{...
- 4,274
5
votes
2
answers
283
views
Closed formula for the asymptotic limit of a definite integral
I would like to solve the following integral:
$$ I_0 (a,b)= \int_0^1 dx\int_0^{1-x} dz \frac{1}{a z (z-1)+a x z + x(1-b)}$$
in the limit where $b$ is small ($a$ and $b$ are positive constants).
...
- 87
9
votes
2
answers
653
views
Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi $
Is there a closed form for $|r|<1$ and $\ell>0$ integer?
The solution for the special cases $\ell=2$ and $4$ would also be interesting if the general case is not available.
Integrating ...
- 1,258