Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
74,716
questions
4
votes
1
answer
63
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Integrate the product of a heaviside step and the absolute value?
I have a rather tricky integral here:
$$\underbrace{\int_0^R r_0\Theta(R-r_0)|r-r_0|dr_0}_{(1)} - \underbrace{\int_0^R r_0\Theta(R-r_0)|r+r_0|dr_0}_{(2)} \ \ \ \ \cases{0\le r < \infty \\ R=1}$$
...
- 583
0
votes
1
answer
48
views
Solving challenging 4D integrals arising from triangle-triangle gravitational interaction
I am trying to find a closed form for two related integrals, coming from a physics problem partially solved here, about attractive forces between two triangles :
$$\begin{align}
{\bf F}_1 &= -G ...
- 125
3
votes
0
answers
37
views
Determining the significance of a curve's factors
Given the equation $x^2+x+1$ you could easily determine that $x^2$ will have the greatest overall impact on the curve--then $x$ and finally $1$. And this holds true for any coefficients present as the ...
- 31
4
votes
2
answers
309
views
Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$
While exploring possible applications for exponential substitution, I stumbled upon the following integral identity:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
- 1,011
4
votes
2
answers
133
views
Reference for $\int_{-\infty}^{\infty}e^{a x^4+b x^3+cx^2}dx\;$?
In my research I encounter an integral of the form
$$
\int_{-\infty}^{\infty}{\rm e}^{\large ax^{4}\ +\ bx^{3}\ +\ cx^{2}}\,{\rm d}x\qquad a < 0,\quad b, c \in \mathbb{R}
$$
So the integral is ...
- 41
-1
votes
1
answer
39
views
Recursive piecewise integral formula [closed]
I have the recursive formula for the integral $1/(x^2+a^2)^n$, which is, in fact, the one that Ng Chung Tak provides in this link. My problem is that when finding a specific integral, for the case $n=...
-2
votes
2
answers
157
views
What is the sign of $I_n = \int _{0}^{1}\frac{x^{2n+1}}{x^{2}+1}dx$ [closed]
I was given an exercice to calculate $I_0$ and then $I_0 + I_1$ and then deduce $I_1$, and then asks the sign of $I_n$, can someone help? I tried deductive reasoning but I don't know how to complete ...
- 15
1
vote
0
answers
56
views
Differentiation under integral signs as done in basic quantum mechanics
In various text books, lectures or lecture notes on basic quantum mechanics, I've seen cases differentiating under integral signs and I am wondering why it is allowed in those situations.
The typical ...
- 2,436
0
votes
1
answer
72
views
Electric field in the plane of a charged ring
It's basically this question Electric field off axis inside a charged ring., but I want to know if it is possible to solve this integral analitically.
\begin{align}
E= \dfrac{k\lambda}{R}\underbrace{\...
- 155
0
votes
0
answers
45
views
Prove/disprove upper bound and lower bound of the Integral
Hey I need to Prove or disprove this sentence:
$$
\frac{4}{9}(e-1) \leq \int_0^1 \frac{e^x}{(1+x)(2-x)} \, dx \leq \frac{1}{2}(e-1)
$$
using the infimum and supremum method for integrals, where m and ...
- 1
0
votes
0
answers
69
views
Landau Notation Problem
I have this function
$$ K_{n} = \int_{1}^{+\infty}\frac{1}{(1+t^2)^n}dt$$ $$ \text{Let }t\geq1,t^2+1\geq1+t\Leftrightarrow\frac{1}{1+t^2}\leq\frac{1}{1+t} \text{ and for } n \in {\mathbb{N^{*}}} : \...
0
votes
2
answers
63
views
Integration of function $ [ \int_{0}^{\pi} |\sin x - \cos x| \, dx ] $ [closed]
Hey I need to evaluate this definite integral:
$
[
\int_{0}^{\pi} |\sin x - \cos x| \, dx
]
$
Don't really know how to approach this, would glad if someone can show me the way to solve this.
I can't ...
- 1
0
votes
0
answers
35
views
Related to double integration
In my research work, I got the following expression:
$\int_0^{\infty} \int_0^{Z_{lim}} 1-\exp(-\lambda*\frac{(\gamma_{th}([A+\beta^2y^2](P_u/jj1)+1/jj1)-\beta^2yz(\alpha_u-\gamma_{th}\alpha_t))}{\...
- 183
4
votes
0
answers
76
views
Validity of Python-derived solution for contour integral $\oint f(z)f(z-\overline{z})~dz$
$\newcommand{\on}[1]{\operatorname{#1}}$
$$
\mbox{Consider the function:}\quad
\on{f}\left(z\right) =
\frac{{\rm e}^{tz}}{\left(1 + z^{2}\right)^{3}}\,
\left(\sqrt{t} - t\right)\ \ni\ t,z \in \mathbb{...
- 41
0
votes
0
answers
25
views
Integral of Poisson Kernel
This doubt comes from Dupaigne's book named stable solutions of elliptic partial differential equations.
The Poisson Kernel is
\begin{equation}
P(x,y)=\frac{\partial G(x,y)}{\partial n_{y}}=\frac{1-|x|...
- 89