All aspects of integration, including the definition of the integral and computing indefinite integrals (antiderivatives).
4
votes
2answers
71 views
A delta function conjecture: almost any function can be a delta kernel
I have been thinking about delta kernels, and I think I have come up with a surprising result:
If $f:\Bbb{R}\to\Bbb{R}$ is such that $\int_{-\infty}^\infty f(x)\,dx=L$ is finite and nonzero, then ...
2
votes
1answer
84 views
$\int_{-\infty}^{\infty}\!e^{- \pi (x+iy)^2}\,dx = 1$ for all $y$.
Can anyone provide a proof of why $\int _{-\infty} ^ {\infty} e^{-\pi (x+iy)^2} dx$ equals 1, for all y ? $x$ and $y$ are real numbers.
EDIT: We already know this for y=0.
Thank you.
2
votes
3answers
83 views
General solution of differential equation of order 3
Please ,how to find that the general solution of $u'''(t)=e(t) , t\in [0,1]$ is given by
$u(t)=c_0+c_1t+c_2 t^2 +\frac12 \int_0^t (t-s)^2 e(s) ds$
$e:(0,1)\rightarrow \mathbb{R}$, and $e\in ...
1
vote
2answers
190 views
+100
Electrostatic Potential Energy
How is the boxed step , physically as well as mathematically justified and correct ?
Source:Wiki http://en.wikipedia.org/wiki/Electric_potential_energy
As work done = $- \Delta U $. for Conservative ...
2
votes
2answers
36 views
Runge-Kutta and Step doubling
I am studying Runge-Kutta and step size control and came up with a few doubts. Because they are related with this integration method, I will divide it in two parts. First, allow me to introduce the ...
0
votes
0answers
42 views
Integrating along a line of points; two different approaches, two different problems
I have an integration question, and I've taken two different approaches which both have serious flaws and don't agree; I'll outline them here and would be grateful for any guidance. I have an infinite ...
1
vote
1answer
28 views
determine whether an integral is positive
Given a standardized normal variable $X\sim N\left(0,1\right)$, and constants $ \kappa \in \left[0,1\right)$ and $\tau \in \mathbb{R}$, I want to sign the following expression:
\begin{equation}
...
3
votes
1answer
63 views
Why is Cauchy's integral formula always written with the function as the subject?
Why is Cauchy's integral formula always written as $$f(w)=\frac1{2\pi i}\int_L\frac{f(z)}{z-w}dz$$ instead of as $$\int_L\frac{f(z)}{z-w}dz=2\pi i f(w)$$? Isn't the latter form how it's typically ...
6
votes
3answers
99 views
Integral of $\cot^2 x$?
How do you find $\int \cot^2 x \, dx$? Please keep this at a calc AB level. Thanks!
5
votes
2answers
136 views
$\int_0^1 \sqrt {\tan^{-1}x}\space dx=$?
How to evaluate $$\int_0^1 \sqrt {\tan^{-1}x}\space dx\qquad ?$$ Is there an elementary expression or value for it? (though I know that there is no elementary expression for $\int \sqrt ...
1
vote
1answer
34 views
Integral involving normal densities
I am trying to solve the integral
$$I(y)=\int_{\mathbb R}f(x,y)g(x)dx,$$
where $f(x,y)$ is the bivariate normal density with known mean $(\mu_1,\mu_2)$ and covariance matrix $\Sigma$ , and $g(x)$ is ...
2
votes
1answer
74 views
Simple conceptual Fundamental Thm of Calculus question
When applying the Fundamental Thm of Calculus in complex analysis, what does it mean for an open connected set to contain a loop? For example, does my red-color open annulus contain the black colour ...
1
vote
1answer
51 views
Integral sign with circle (AND arrow on the circle) through it
I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path.
So when I encountered in complex analysis the above integral sign but with ...
2
votes
3answers
96 views
Vector calculus for ellipse in polar coordinates
I'm having trouble with this question, can somebody please help me with it! I'll thanks/like your comment if help me =)
I know that for a ellipse the parametric is $x=a\sin t$ , $b= b \cos t$, ...
6
votes
1answer
76 views
What is the relation of $\int f dx^1\wedge dx^2\wedge …\wedge dx^n=\int f dx^1…dx^n$
In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function ...
1
vote
1answer
56 views
Show that a function is continuous
Let K be bounded and continuous and bounded on $\mathbb{R}^{n}$ and let $f$ be Lebesgue integrable on $\mathbb{R}^{n}$.
Show that the function $g$ defined on $\mathbb{R}$ by
$g(t) = ...
0
votes
0answers
25 views
Conditional expectations calculation, check my work please.
Let $f_{X,Y}(x,y)=2(x+y)$, for $0<y<x<1$. Find $E[X|Y], E[Y|X]$.
This is purely a calculation, but that's my weakest spot, I always make some stupid mistake that loses me half the points! ...
2
votes
1answer
52 views
Calculate the Riemann Stieltjes integral
This is not a homework question. It is a past exam question and I would appreciate some step by step help, as I never understood this concept in class.
Let $\alpha(t) = n^2$ for $t\in[n,n+1).$ ...
4
votes
1answer
123 views
Is line integration a generalization of the definite integral in $\mathbb{R}$?
Recently I've been writing integrals in the following way, for example
$$\int\limits_{[0,1]} {{t^{y - 1}}{{\left( {1 - t} \right)}^{x - 1}}dt} $$
instead of
$$\int\limits_0^1 {{t^{y - 1}}{{\left( ...
0
votes
1answer
15 views
Analysis: Integration (Riemann/Step functions)
Using the definition of the integral of a continuous function, and
that $\displaystyle\sum_{j=0}^{2^n-1} j = (2^n-1)2^{n-1}$ to show that
$\int_0^1x \ dx = \frac{1}{2}$
I'm having trouble ...
0
votes
1answer
136 views
Manipulating a product term inside an integral
I have an expression of the form
$$P(x) = \int_0^\infty \prod_{i=0}^{n} e^{-G(a, x_i)}\,\mathrm{d}a $$
and I was wondering if there was any way that I could swap the order of the product and the ...
1
vote
1answer
28 views
Integration of volumes of revolution: bisector surface
$y = (16-x^2)^{0.5}$ is rotated around the $x$-axis to give a sphere of radius $4$ units. Find the equation of the straight line that passes through $(-4,0)$, such that when also rotated around the ...
0
votes
1answer
27 views
Question on integration
Let $(p,q)\in \mathbb{R}^2$ , and $H: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$
if $q'=\frac{\partial H}{\partial p} (p,q) ~~\text{and}~~ p'=-\frac{\partial H}{\partial q} (p,q)$
How to ...
1
vote
0answers
76 views
The integral of $\frac{1}{1+x^n}$
Motivated by this question:
Integration of $\displaystyle \int\frac{1}{1+x^8}dx$
I got curious about finding a general expression for the integral $\int \frac{1}{1+x^n},\,n \geq 1$. By factoring ...
15
votes
4answers
1k views
Why doesn't integrating the area of the square give the volume of the cube?
I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume):
$$V = \int\limits_a^b {A(x)dx}$$
But this doesn't seem to work with the square. ...
3
votes
1answer
239 views
Discontinuity in function; order of integration matters
I'm struggling a bit with Chapter 10 of Rudin's "Principles of Mathematical Analysis," and I was hoping to get some help here.
I'll post the problem and my current progress.
Exercise 2: For $i = 1, ...
1
vote
1answer
66 views
Show that $\int f(x)dx=\int g(u)du$
Dose anyone know how to do this?
If a function $f(x)$ can be written $f(x)=g(u(x))du(x)/dx$ for a suitable function $u(x)$, show that $\int f(x)dx=\int g(u)du$.
0
votes
3answers
487 views
Volume of half torus. What's wrong with my solution?
A doughnut has been partially eaten by a meticulous person so that the portion remaining is given by rotating the half-circular region shown above about the y axis. What fraction of the original ...
2
votes
2answers
37 views
Evaluate $\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$
I'm trying to evaluate the integral $$\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$$
where $\chi_{[0,1]}(x)=1$ is the characteristic function, i.e. equals $1$ for $x \in ...
3
votes
1answer
44 views
Using a definite integral, to create a specific recurrence relation.
Hello i have the integral:
$$y_n=\int_0^1\frac{x^n}{x+5}dx$$ where $ n=1,2,3,4,....,\infty$
I need to show that the integral can be represented by the recurrence relation below;
$$y_n= ...
4
votes
3answers
90 views
Integral of rational functions.
I want to evaluate this integral:
$$\int{\frac{ax+b}{(x^2+2px+q)^n}}dx$$
The book only says to integrate by parts $\int{\dfrac{1}{(x^2+2px+q)^{n-1}}dx}$,
for simplicity if $n = 2$ I get:
...
1
vote
0answers
49 views
Antiderivative of an absolute function
$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$
$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to ...
2
votes
1answer
39 views
Leibniz rule, multiple integrals
Suppose I need to compute the derivative
$$
\frac{d}{dr} \int_{-\infty}^{\infty} \int_{h(r)}^\infty \int_{g(r)}^\infty {rf(x,y,z)\, dz\, dy\, dx}.
$$
Can I apply a Leibniz rule of some form? How?
2
votes
3answers
82 views
$\int^{\pi/2}_{0}\log|\sin x| \,dx = \int^{\pi/2}_{0}\log|\cos x| \,dx $
Prove that :
$$\int^{\pi/2}_0 \log|\sin x| \,dx = \int^{\pi/2}_0 \log|\cos x| \,dx $$
I tried to cut the integral into a sum of parts and changing variable but it didn't work out right, i dont ...
1
vote
1answer
52 views
How to find the unknown values in this Numerical Integration type?
Given the following type of numerical integration:
$$I(f)=\int_0^1 f(x) \, dx \approx \frac 12 f(x_{0}) +c_1 f(x_1) $$
a) Find the values of: the coefficient $c_1$ and points $x_0$ and $x_1$ so ...
12
votes
4answers
281 views
Which methods to use to integrate $\int{\frac{x^4 + 1}{x^2 +1}}\, dx$
I have this integral to evaluate:
$$\int{\frac{x^4 + 1}{x^2 +1}}\, dx$$
I have tried substitution, trig identity and integration by parts but I'm just going round in circles.
Can anyone explain ...
11
votes
2answers
168 views
Evaluating $ \int^{\infty}_{-\infty}\sin\left({\pi}^{4}x^{2}+\frac{1}{x^2}\right) dx$
$$\int^{\infty}_{-\infty}\sin\left({\pi}^{4}x^{2}+\frac{1}{x^2}\right) dx$$
This is a problem from the Pi Mu Epsilon Journal, and I'm having great trouble answering it. I've tried some substitutions ...
1
vote
1answer
67 views
A little help integrating this torus?
Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by
$$\mathbf{F}(x,y,z)=(x,y,z).$$
Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
1
vote
1answer
41 views
Inverse Laplace Transform. Computing the integral.
This question is related to this one, but I'm hereby taking a different approach.
Problem: Solve
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their ...
0
votes
1answer
34 views
Riemann integral show $f(x)=g(x)$ for at least 1 $x$ in [a,b]
Let $f$ and $g$ be continuous functions on $[a,b]$ such that $\int_a^b f = \int_a^b g$. Show that there exists $x\in [a,b]$ such that $f(x) = g(x) $.
I want to assume not and then show that the ...
1
vote
1answer
34 views
Monotonic integral proof
Let $f$ be a continuous function on $[a,b]$ such that $f(x) \geq 0 $
for every $x\in [a,b]$. Suppose $\int_a^b f = 0$ and show that $f (x) = 0$ for every $x\in [a,b]$.
obv this is monotonic ( ...
2
votes
1answer
130 views
Help solving the following intergral $\int_{-\infty}^{\infty}\frac{1}{\sqrt{k-p}\sqrt{k+p}}dp$
I was wondering about calculating the following definite integral analytically:
\begin{equation}
\int_{-\infty}^{\infty}\frac{1}{\sqrt{k-p}\sqrt{k+p}}dp
\end{equation}
Does someone know how to ...
3
votes
0answers
30 views
Stokes' Theorem and Measure Zero Sets
This is probably a very naive question but I am trying to connect two pieces of information in my head regarding integration of differential forms and integration with respect to a measure.
The first ...
1
vote
2answers
61 views
Evaluate $\int \dfrac{1}{\sqrt{1-x}}\,dx$
Find $$\int \dfrac{1}{\sqrt{1-x}}\,dx$$
I did this and got $\dfrac23(1-x)^{\frac32} + c$
But a online calculator is telling me it should be $-2(1-x)^{\frac12}$
What one is on the money and if not ...
6
votes
4answers
279 views
What technique would be suitable to solve this: $\int \sin ^{5}\left( x^{2}\right) \left( x\cos \left(x^{2}\right)\right)\mathrm{d}x$
I think integration by parts might work but I'm now sure. Thanks very much.
2
votes
3answers
46 views
Find the antiderivative of $\sqrt{3x-1} dx$
Find the antiderivative of $\sqrt{3x-1} dx$.
I got $\frac{2}{3}(3x-1)^{3/2}+c$ but my book is saying $\frac{2}{9}(3x-1)^{3/2}+c$
Can some one please tell me where the $2/9$ comes from?
1
vote
1answer
29 views
Question about integrability
Let f be a continious function on [a,b] and exist a partition P of [a,b] such that $\bar{S}(f,p)=\int_a^b f(x)dx$. Prove that f is a constant function. I thought stratting assuming the claim is not ...
8
votes
2answers
100 views
Evaluating $\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$
I would like to evaluate in a closed form the integral
$$\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$$
I tried various methods :
integration by parts
some changes of variables ...
1
vote
3answers
62 views
Is this an exact differential or not?
I have the 1-form
$$dz=2xy\, dx+(x^{2}+2y)\, dy$$
And I want to integrate it from $(x_{1},y_{1})$, to $(x_{2},y_{2})$.
If I'm not drunk, checking mixed partials, I find that $dz$ is an exact ...
2
votes
2answers
64 views
evaluate $\int\ln x\tan x\,dx$
How to evaluate $\int\ln x\tan x\,dx$ ?
I've tried to do integration by parts but after calculations it cancel out the main question.
