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.
74,709
questions
-5
votes
0
answers
54
views
A simple yet complex proof that I am unable to solve. [closed]
prove that: ((e^(ax ))cos(bx))^n=((sqrt(a^2+b^2))^n)(e^(ax))cos(bx + n.arctan(b/a))
0
votes
0
answers
78
views
Looking for citation of a definition of multivariable integral
I need a definition of Multivariable Riemann Integral to cite in my article.
I've been searching different cited sources, for example Rudin, Folland, Stewart and Larson books.
In the first two I ...
0
votes
0
answers
49
views
Am I correctly applying repeated integration by parts?
Say we have two compactly supported functions $f,g:\mathbb{R}^n\to\mathbb{R}$. I found myself computing
\begin{equation}
\begin{split}
\int_{\mathbb{R}^n}f\frac{\partial^{r}g}{\partial x^{\alpha_1}\...
- 5,166
-2
votes
1
answer
106
views
Is there an analog for factorials in division, and if so, what are its applications and properties? [closed]
If we consider a factorial to be an operation/function of iterative multiplication, would it be reasonable to think that something similar for division also exists?
If we take this function to be f, ...
0
votes
1
answer
54
views
Volume under a normal distribution in cartesian and polar not integrating to 1
This is my first calculus problem in several years as a professional. I'm checking to see that my integral of a normal distribution works, so that I can use it as a weighting function for ...
0
votes
0
answers
17
views
Integration of the product of a compact supported convolution [closed]
I know that in general case we have $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(s)g(t-s) ds dt = \left( \int_{-\infty}^{+\infty} f(t) dt \right) \left( \int_{-\infty}^{+\infty}g(t) dt \right) ...
- 13
2
votes
2
answers
67
views
Integral of exponential function multiplied by sine function [duplicate]
Question: Find $\displaystyle\int e^x\sin x\,dx$
Let $I = \displaystyle\int e^x\sin x\,dx$
Then
$$
I = e^x (-\cos x) - \int\bigl[e^x(-\cos x)\bigr]\,dx
$$
Integrating $e^x(-\cos x)$:
$$
I = -e^x\cos ...
- 199
0
votes
2
answers
53
views
Integral of a Generalized Laguerre Polynomial [closed]
I am looking for the solutions to the following integral:
$$
I_{n} =
\int_{0}^{\infty}x^{4}
\operatorname{L}_{n}^{3}\left(x\right)
{\rm e}^{-\left(n + 3\right)x/2}\,{\rm d}x,\qquad n \in\mathbb{N}_{0}
...
- 19
0
votes
0
answers
14
views
moment of inertia of cylinder of variable density + uniform rod - Part b).
The density at a distance $r$ from the axis of a circular cylinder, of mass $M$ and radius $ a $, is $ \frac{ \rho_{0}a}{r} $, where $ \rho_{0} $ is a constant.
a). Prove by integration that the ...
- 79
2
votes
1
answer
76
views
Double integral of $x^2y+y \sin(x^9)$ dxdy
I have some troubles with the following double integral (in particular the part with sinus)
$$
\iint_{D}\left[x^{2}y + y\sin\left(x^{9}\right)\right]{\rm d}x\,{\rm d}y\quad
\mbox{where}\quad D\ \mbox{...
2
votes
2
answers
149
views
Evaluating $I=\int_1^\infty \frac{2(z^2-1)}{(z^2+1)^2\ln(z)}\,dz$
$$
I =
\int_{0}^{\infty} \frac{\tanh\left(x\right)\operatorname{sech}\left(x\right)}{x}\,{\rm d}x
$$
I started with the substitution $\cosh x=t \implies [0,\infty] \to[1,\infty]$
$$I=\int_1^\infty \...
- 2,879
3
votes
1
answer
98
views
Contour integration of an integral
I am trying to determine the following integral by using the residue theorem
$$\int_{ - \infty }^\infty {\frac{{\gamma \left( {1/2,{\rm{i}}z} \right)\gamma \left( {1/2, - {\rm{i}}z} \right)}}{{{z^2} +...
- 31
0
votes
1
answer
31
views
Integration with respect to cdf.
Let $F$ be a continuous and strictly increasing CDF such that $F(x) \neq 1$ for all $x \in \mathbb{R}$. Let $U(x) = F^{-1}(1-\frac{1}{x})$.
The claim is that for $n \in \mathbb{N}$, $z \colon \mathbb{...
- 1,710
0
votes
1
answer
21
views
Measurability of a family of parametric integrals assuming measurability of the integrand w.r.t. the parameter
Let $D$ and $E$ be measurable subsets of $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, and $v : (x,t) \in D \times E \mapsto v(x,t) \in \mathbb{C}$.
Assume that the maps $v(\cdot, t)$, $t \in E$ ...
- 5,849
1
vote
0
answers
44
views
Special case of sifting property of dirac-delta at singularity.
Note: This is not a duplicate. You can see my linked old question below in the text and realize that my question is different.
I have an integral of the following form:
$$
I = \lim_{a\ \to\ \infty} \...
- 862