Questions tagged [definite-integrals]
Questions about the evaluation of specific definite integrals.
21,712 questions
10
votes
1
answer
211
views
Definite integral with the Lambert W Function
I am trying to find the integral
$$
\int^{1}_{0} \frac{\mathrm{W} \left( -x\log x\right) }{-x\log x} \mathrm{d} x=\frac{\pi^{2} }{12}
$$
By expanding the Lambert W-Function into a series, one ...
- 1,179
-1
votes
0
answers
39
views
Solving a power tower integral [duplicate]
Studying a special integral, namely
$$
I:=\int^{1}_{0} \left( x^{x}\right)^{\left( x^{x}\right)^{...} } \mathrm{d} x
$$
Note that the expression
$$
y:=\left( x^{x}\right)^{\left( x^{x}\right)^{...} ...
- 1,179
15
votes
1
answer
279
views
Trying to show that $\int_{0}^{\infty}(a^{-x})^{(a^{x})^{(a^{-x})^{(a^{x})^{\cdots}}}}\,dx=\frac{\pi^2}{12\ln a}$
I was creating different power tower integrals and stumbled upon this. Numerically (for $a>1$), it seems like
$$\int_{0}^{\infty}(a^{-x})^{(a^{x})^{(a^{-x})^{(a^{x})^{\cdots}}}}\,dx=\frac{\pi^2}{12\...
- 153
2
votes
1
answer
134
views
Evaluate $\int_{0}^{\infty} \Gamma(1+it) (it)^{-it} dt$
For context, my original attempt was at the integral $\int_0^\infty \frac{x!}{x^x}dx$. However, I ended up having to evaluate the integral in the title.
Let's begin by naming our complex function $f(z)...
- 308
4
votes
1
answer
104
views
Closed-form or special-function expression for this integral
I would like to obtain an expression to evaluate, without numerical integration, the following integral:
$$I(a,b,m)=\int_{1}^{c>1} (x^{2}-1)^{\frac{m-1}{2}}\;x^{-1/2}\;
e^{a\sqrt{x^{2}-1}-b x}\,dx$$...
- 165
0
votes
0
answers
33
views
In my proof, did I handle order of intermediate points and $g_{2}(x)$ properly?Parts I have doubts are enclosed in block quotes.
The unverified proof please refer to problem and proof
.In the proof,I am uncertain about accuracy of two sections.
Explanation of rationale for defining $g_{2}(x) $ after Selecting $\eta _{1}$:Since ...
- 29
-1
votes
0
answers
45
views
How to solve $\int_0^{\pi/2}e^{-\frac{\pi}{2}\tan(x)}dx$? [duplicate]
How to solve $$\int_0^{\pi/2}e^{-\frac{\pi}{2}\tan(x)}dx$$? I tried differentiating under the integral and contour integration but couldn't find anything useful
1
vote
1
answer
127
views
Is there an elementary way to evaluate $\int_0^1 x\, e^{\frac{x^2 - 1}{2}} \cos x\, dx$? [closed]
$$
\int_0^1 x\, e^{\frac{x^2 - 1}{2}} \cos x\, dx
$$
I tried standard techniques like integration by parts and substitution for example $u=\frac{x^2}{2}$, but I couldn't find a way to evaluate this ...
- 11
8
votes
4
answers
271
views
Help in the elliptic integral $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}x \csc^2x\sqrt{4-\csc^4x}dx$.
Someone asked me a question about integrals.
$$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}x \csc^2x\sqrt{4-\csc^4x}dx$$
I broke it down into several integrals using integration by parts.
enter image ...
- 81
4
votes
1
answer
97
views
Solving the double integral $\int_0^{2\pi}\int_0^a \frac{z_0 \rho}{(\rho^2 + \rho_0^2+z_0^2 -2\rho \rho_0 \cos \phi)^{3/2}} d\rho d\phi$
I came across this integral while solving for the axial component of the field due to a uniformly charged ($\sigma_S$ is charge density) thin disk at an off-axis point P. If $k_E = \frac{1}{4\pi \...
- 1,318
3
votes
2
answers
134
views
Closed-form or asymptotic approximation for integrals involving $\sqrt{\log t}$ and $\frac{1}{\sqrt{\log t}}$
I'm working on a number theory research project where I'm analyzing the distribution of primes expressible as the sum of two squares. A heuristic approximation I derived for the counting function ...
- 143
2
votes
1
answer
104
views
Evaluating $\int_0^1 \frac{\left(1-u^2\right)^2 (1+u^2)}{u^6}\,\exp\left( \sigma\left( u-\frac{1}{u} \right)\right)du$ with $\Re\{\sigma\}>0$.
While working on a fluid mechanics problem involving a porous medium, I encountered the following definite integral:
$$
f(\sigma) = \int_0^1 \frac{\left(1-u^2\right)^2 (1+u^2)}{u^6} \, \exp \left( \...
- 486
1
vote
0
answers
171
views
Is there a known closed form of $\int_0^1 \frac{\mathrm{Li}_5(-x^2)}{1+x}\,dx$?
I have found 2 closed forms of some lower versions of these integral on Instagram, I verified them on Wolfram and they were correct, now I am curious to find the closed form for the last integral on ...
user1660564
2
votes
1
answer
103
views
Evaluate $\frac{2}{\pi} \int_{0}^{\pi} du \, \arctan\left[\frac{1}{a}\cos(u)\right]\, \cos\left[(2n+1) u\right]$ without contour integration
The question and my attempted solution
Show
\begin{eqnarray*}
\frac{2}{\pi} \int_{0}^{\pi} du \, \arctan\left[\frac{1}{a}\cos(u)\right]\, \cos\left[(2n+1) u\right] & = & (-1)^n \frac{2\, \...
- 63
2
votes
1
answer
111
views
$\int_{m}^{n} \sin(\pi x)(rx)^{2k-1} dx$ > $\int_{m}^{n} \sin(\pi x)(rx)^{2k+1} dx$
I am struggling with the following problem:
Suppose that $0 < m < n < \frac{1}{r}$, where $r$ is a small positive constant, $m$ is an even integer, and $n$ is an odd integer.
Let $I_k = \int_{...
