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.
74,490
questions
-1
votes
0
answers
19
views
Find a recurrence relation for a polynomial integral [duplicate]
Its expression is:
$$\int_0^1{(x-x^2)^n}, dx$$
0
votes
0
answers
22
views
Integrating a function composed with a Dirac delta [closed]
In a physics problem I found an integral of the form:
$$
\int d^4x\exp\left[i\delta(t-\tau)\hat{A}\right],
$$
where $\hat{A}$ is an hermitian operator, $\tau\in\mathbb{R}$ and $d^4x=dtd^3x$. The ...
- 121
2
votes
1
answer
49
views
How to Solve $\int \frac{x + 1}{(x^2 + 6x + 14)^3} dx$?
I am trying to solve the following integral and would appreciate some guidance:
$$\int \frac{x + 1}{(x^2 + 6x + 14)^3} \, dx$$
I have attempted various methods, including substitution and integration ...
- 31
0
votes
2
answers
51
views
Complex integrals that look like they agree, differ by sign (according to Mathematica)
Consider the integral
$$\int_0^\infty \frac{dz}{1-z^2 +i0^+},$$
I would assume it to agree with the integral
$$\int_0^\infty \frac{dz}{(1-z+i0^+)(1+z+i0^+)}. $$
However, according to Mathematica the ...
0
votes
0
answers
49
views
Double integral $ \iint_D (x^4-y^4) dx\,dy$
I have troubles with the following integral
$$
\iint_D (x^4-y^4) dx\,dy
$$
over D: $1<x^2-y^2<4, \sqrt{17}<x^2+y^2<5, x<0, y>0$
This is the same problem as in Compute $\iint_D (x^4-y^...
-3
votes
0
answers
40
views
Arc length derivation [closed]
enter image description here
Why does the hypothetical X value Xi that would equal the avg slope of a section get replaced by X? I don't remember using the mean value theorem in the definition of the ...
-1
votes
0
answers
13
views
Convergence of the exponentially-weighted moving average in integral form
Let $f:[0,+\infty)\to\mathbb{R}$ be continuous and let $\bar{f}\in\mathbb{R}$ be such that $f(t)\to\bar{f}$ as $t\to+\infty$. Does this suffice for
\begin{equation}
\int_{0}^{t}{\rm e}^{-\rho(t-s)}f(...
- 301
0
votes
0
answers
28
views
Moments of Pearcy type integral
In my research I encounter Moments of Pearcy Integral
which can be written as
$$
\int_{-\infty}^{\infty}x^{n}
{\rm e}^{-ax^{4} + bx^{2} + cx}\,{\rm d}x\qquad
a > ...
- 41
2
votes
0
answers
60
views
Constant term in the Euler-Maclaurin expansion of $s_n=\sum_{k=1}^n \tfrac{1}{k+1/2}$
This question is a followup from my previous post based on the Euler-Maclaurin formula: How to find the correct constant term with Euler-Maclaurin formula, $\sum_{j=1}^n j\log j$.
This time I am ...
- 2,247
0
votes
0
answers
46
views
Approximation of a Riemann sum.
Given a twice continuously differentiable function $f\in C^2([0,1])$, is there a theorem/result/algorithm on how to place $0<x_1<\ldots<x_{n-1}<1$ so that adding $x_0=0$ and $x_n=1$,
$$
\...
- 208
0
votes
0
answers
39
views
$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$.Calculate $\int_A xyz \ d \lambda_3$. I need to verify my solution.
$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$
Calculate:
$$\int_A xyz \ d \lambda_3$$
Solution:
We know that: $x^2 + y^2 + z^2 > 0$ and therefore $2x + 2y > 0 \iff x + y > 0$
...
- 1,076
0
votes
0
answers
40
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Conditions for function to be periodic
I am investigating the following type of functions
\begin{equation}
I(\alpha) = \int_{0}^{\pi}f(t)\cos(\alpha t)\,\mathrm{d}t\,.
\end{equation}
where $f(t)$ is a real-valued function with non-negative ...
- 183
0
votes
0
answers
24
views
Will the following Method of engineering analysis work?
Analytical Engineering Analysis of 3D Shapes
Using volume integral($\iiint_{}^{}{f(t)}dx dy dz$) to do a Analytical
Engineering Analysis of 3D Shapes without using mesh based FEA. Like
integration ...
11
votes
6
answers
1k
views
Formula for bump function
I would like to formulate a bump function (link) $f(x)$ with the following properties on the reals:
$$
f(x) :=
\begin{cases}
0, & \mbox{if } x \le -1 \\
1, & \mbox{if } x = 0 \\
0, & \...
1
vote
2
answers
33
views
Laplace transform of $\sin(\omega t)$
I am learning about the Laplace transform and I know I got the answer to this example question wrong, but I'm trying to figure out if I just made a calculus or algebra type error, or if I'm ...
- 125