All Questions
Tagged with mathematica definite-integrals
20
questions
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Definite Integral involving polynomial and Meijer G-function
I am trying to solve the following integral involving Meijer-g function
$\int_0^{\infty} \left(\varrho x + \nu\right)^{p+\frac{1}{2}-\frac{k}{2}} G_{1,3}^{3,0}\left(\begin{matrix} \frac{\...
- 31
1
vote
0
answers
41
views
regularized incomplete beta function integration
Solve $\int_{0}^{1}\frac{I_{u^{\frac{1}{p}}}\left ( p+\frac{1}{a} ,1-\frac{1}{a}\right )}{u}du$ . In Mathematica, this integral does not converge but from an article, I got the answer to this integral ...
0
votes
0
answers
42
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How to proceed with this integral in Mathematica?
NIntegrate[
Cos[2 1 Pi (2 Sqrt[(n + 8854800957672623)^2 -
78407500000000000000000000000000] + 0) -
Pi]/(2 1 Pi)^2 (1000000/(10^40 Pi^8 (Sqrt[
2 (Sqrt[(n + 8854800957672623)^2 -
...
- 77
1
vote
2
answers
72
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Having trouble with the integral of the second derivative of the function $e^{\frac{-1}{1 - x^2}}$ after a change of variables
The first thing I have to note is that I am not 100% sure whether this problem is due to Mathematica or just the mathematics I have produced. Thus feel free to direct me to the appropriate forum.
Let $...
- 1,030
4
votes
0
answers
79
views
Closed form of A Log-Sine-Cosine Integral by Mathematica
By using the Mathematica 12.0, I found the numerical value of a Log-Sine-Cosine Integral that
$$\int_0^{\frac{\pi}{6}} \log^2(2\sin\theta)\log(\tan(\theta/2))d\theta=-4.39530911884935704937076725879.$$...
- 51
0
votes
1
answer
102
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Numerically integrating $\int_{20}^\infty\frac{4^{1+4ix}\Gamma(-4ix)e^{-2ix}}{(-2i)^{-4ix}}dx$
I want to compute the following integral numerically in Mathematica,
$$\int_{20}^{\infty} \frac{4^{1+4ix}\Gamma(-4ix)e^{-2ix}}{(-2i)^{-4ix}}dx$$
The problem is that when evaluate the integral from $20$...
- 31
7
votes
3
answers
229
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General formula for $\int^1_0 x^\alpha \log(1-x)\operatorname{Li}_2 (x)\, \mathrm dx$
Consider a following definite integral
$$I(\alpha) = \int^1_0 x^\alpha \log(1-x)\operatorname{Li}_2 (x)\, \mathrm dx$$
Mathematica is able to provide a result for many $\alpha$ integers. See table ...
- 2,903
2
votes
1
answer
117
views
How did the author solve this integral?
I have the following spherical density distribution:
$\rho(x, z) = \frac{1}{\sqrt{x^2 + z^2}\left(1+\sqrt{x^2+z^2}\right)^2}$
which I have broken into a "line of sight" dimension $z$ and a &...
- 339
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votes
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answers
36
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Integrate special function
I have the following integral
$$ \int_a^b e^{\alpha_1 x + \alpha_2 x^2 + \alpha_3 e^{-\beta x}} dx, \\ \text{where} \, \alpha_1, \alpha_2, \alpha_3 \,\text{and} \, \beta \, \text{are constants} $$
I´...
2
votes
1
answer
115
views
Understanding Mathematica's formula for $ \int_0^{\infty } x^a \exp \left(-\frac{c x^2+f x}{b}\right) \, dx $
My goal is to integrate the following function:
$$
\int_0^{\infty } x^a \exp \left(-\frac{c x^2+f x}{b}\right) \, dx
$$
where, $a, b, c > 0$ and $a, b, c, f \in \mathbb{R}$.
Mathematica gives me ...
- 157
0
votes
0
answers
38
views
Length of the arc traveled by a particle with some launch conditions
I have a 3D particle that is launching with some initial position, velocity, and with a constant force. I can solve for it's position at some time with the following equation:
f
[
...
4
votes
1
answer
224
views
May someone of you provide me (if possible) an expression for the F.T. of $\tan(x)$?
So here it is my problem: I need the Fourier Transform of the tangent function, namely:
$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \tan(x)\ e^{-ikx}\ dx$$
I tried for hours by hands and I got ...
- 26.3k
3
votes
1
answer
121
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Integral over fraction of trigonometric functions
I would like to calculate the following integral
$$
I_1(A,B,C) = \int_0^{2\pi} \frac{d\phi}{1 + A \cos\phi + B \sin\phi + C},
$$
and if possible also
$$
I_2(A,B,C,D,E) = \int_0^{2\pi} d\phi \frac{D ...
- 858
4
votes
1
answer
672
views
Integration over the Marchenko-Pastur distribution
Problem Statement:
I want to find closed form expression of the following definite integral
\begin{equation}
\int_{\alpha_{-}}^{\alpha_{+}} \ln x \, \frac{1}{2\pi \alpha x} \sqrt{ (\alpha_{+}- x )(x-...
- 1,005
3
votes
0
answers
66
views
Integrating a function times an exponential of a function plus an exponential yields hypergeometric result
During my studies I came across the following integral which is solvable by Mathematica. I tried to solve it by hand, but have not yet found a trick to get the desired result.
To make it even worse, ...
- 65