Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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2
votes
1answer
59 views

Simple Integral Involving Radicals: Why Does Mathematica Fail?

I have $$\int_{d-1}^{3}\textrm{d}x\left(3-x\right)^3 \sqrt{\left(\frac{2(x-1)}{x}\right) \left(x-\left(d-1\right)\right)}$$ but despite this looking like a simple integral involving fractional ...
1
vote
0answers
62 views

Volume of figure between $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ if $z\geq 0$

I have a problem where I have to find volume of figure formed, when $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ intersects if $z\geq 0$. Here is a graphic for clarity: So far I have transformed the problem to ...
0
votes
1answer
18 views

Separation of variables and why integration of 1/x terms gives ln|x|

So assuming I got something like $$x'(u)=-\frac{x}{u}$$ which gives me then (with separation of variables) $$\int\frac{dx}{x}=\int{-\frac{du}{u}}$$ So my question is: Why do I get $$ln|x|=-ln(u)+c, c ...
1
vote
1answer
32 views

Show that integral of Gaussian distribution is 1

Under a normal distribution, μ = 0 and σ = 1, but when then integrating this equation, I get an error function. Without using Riemann sums, how can I prove that this equation = 1? I have only had ...
0
votes
3answers
38 views

Integral $ \int \frac{1}{x^{1+a} (1-x)^{1-a}} dx~,~a \gt 0$

The following integral is part of a large problem I'm trying to solve and I'm stuck. I'd appreciate some guidance. I would like to know how to compute integrals of the form $$ \int ...
2
votes
2answers
272 views

Find $\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$

It's asked to solve this: $$\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$$ And I have no idea how to do it...
0
votes
2answers
65 views

How to evaluate the integral $\int x^2/\sqrt{4-x^2}\,dx$?

How to compute this integral? $$\int \frac{x^2}{\sqrt{4-x^2}}dx$$ If there were $x$ instead of $x^2$ in the numerator I know how to do a substitution $y=4-x^2$. But this doesn't help with the $x^2$.
0
votes
0answers
30 views

Why is this integral involving the mean value function zero?

Let $u$ and $v$ belong to $H^1(\Omega \times (0,\infty))$ on a bounded domain $\Omega$. Define $$(Au)(y) := \frac{1}{|\Omega|}\int_\Omega u(x,y)\;\mathrm{d}x.$$ We have that $Au \in H^1(0,\infty)$. ...
5
votes
2answers
84 views

Limit $I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$

Im a new participant in this mathematical forum, so this is one of that i couldn't solve it. $$I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$$ I've ...
0
votes
0answers
10 views

Approximating the Arc Length of a Regular Curve with a Broken Line

Question: Suppose $\alpha:[a,b]\to\mathbb{R}^3$ is a regular curve segment. Prove that, for every $\epsilon>0$, there exists $\delta>0$ such that, for any partition ...
0
votes
0answers
32 views

Importance Sampling of 2D constant piecewise function convertible to 1D?

So I have a constant piecewise 2D function (luminance values of pixels of an image) that I am writing an importance sampling algorithm for. I was going to write my algorithm by first sampling the 1D ...
2
votes
1answer
55 views

Evaluating $\int_0^\infty dn \, \frac{x^n}{(3n+1)(3n+2)}$

I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate, $$\int_0^\infty dn \, ...
-1
votes
0answers
9 views

Calculate the expected value of X when $F(x) = \frac12 + \frac1{\pi} \arcsin x$

Given that X is a continuous random variable and its probability distribution function is $$F(x)= \begin{cases} 0, & x\le -1, \\ \frac12+\frac1{\pi}\arcsin x, & -1 \le x < 1, \\ 1, & x ...
1
vote
1answer
32 views

Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
2
votes
2answers
46 views

Prove that there exists $x_0\in [a,b]$ such that $ \sum_{i=1}^{n} k_i \int_{x_0}^{x_i} fdt=0$

Let $f$ is a continuous function on $[a,b]$, $x_1,x_2,\ldots,x_n\in [a,b]$, $k_1,k_2,\ldots,k_n>0$. Prove that there exists $x_0\in [a,b]$ such that $$k_1\displaystyle \int_{x_0}^{x_1} ...
1
vote
0answers
23 views

Integral of symmetric function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be such that $f(x_1,\dots,x_n)=f(x_{\sigma(1)},\dots,x_{\sigma(n)})$ for every $n$-permutation $\sigma$, and suppose that ...
2
votes
1answer
45 views

Integral inequality $\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$

Let $f(0) = 0$ and $0<f'(x)\leq1$ for all $x \geq0$, then prove: $$\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$$ The hint I was given was "differentiate, factor and ...
1
vote
1answer
39 views

Integration of a function defined by its graph, the union of semi-circles and a line segment

I don't understand how to do this problem and I would someone to help me with it.Please step by step for me. I just started on integration so this problem is a bit too hard for me due to my lack of ...
0
votes
0answers
23 views

Change of variable of integration, for numerical integration

I have an independent (array) variable $r = {r_0, r_1, ..., r_N}$, and three functions (arrays) of that variables, $n(r) ={n_0, n_1, ..., n_N}$, $p(r)$, and $E(r)$. How can I calculate the function ...
1
vote
1answer
38 views

How to integrate $12x^3(3x^4+4)^4 $ in a nice way

How would I antidifferentiate $12x^3(3x^4+4)^4 dx$ ? I guess it is possible to multiply it all out, and then do term by term, but is there a more efficient solution?
0
votes
1answer
38 views

Can someone help me with this fundamental theorem of calculus problem dealing with integration by graph? [on hold]

I don't understand how to do this problem and I would like someone to guide me step by step.
0
votes
0answers
27 views

How to prove that the integral of a positive, continuous function is positive?

Obviously intuitively the area under something that is above the x-axis is always positive, but how can I show this with a proof?
2
votes
1answer
45 views

Equivalent of $\int_0^{\pi/2}\cos^n(\sin(x))dx$

Let $\displaystyle u_n=\int_0^{\pi/2}\cos^n(\sin(x))dx$. How can I find an equivalent of $u_n$ when $n\to\infty$ ?
1
vote
3answers
53 views

Is it possible to have simultaneously $\int_I(f(x)-\text{sin} x)^2 dx\leq \frac{4}{9}$ and $\int_I(f(x)-\text{cos} x)^2 dx\leq \frac{1}{9}$?

Let $I=[0,\pi]$ and $f\in L^2(I)$. Is it possible to have simultaneously $\int_I(f(x)-\sin x)^2 dx\leq \frac{4}{9}$ and $\int_I(f(x)-\cos x)^2 dx\leq \frac{1}{9}$? I don't understand what this ...
-1
votes
1answer
32 views

Two Definitions of Lebesgue Integral

So the definition of Lebesgue integral as I understand it is as follows: Let $(X, \mathcal{F}, \mu)$ be a measure space, and $f: X \to [0, + \infty]$ a non-negative function. Then for simple ...
0
votes
4answers
41 views

Given $\int_0^x (x-t+1)g(t)\,\mathrm{d}t = x^4 + x^2,$ Find $g(x)$

(Stanford Math Tournament 2012 #7) A differentiable function $g$ satisfies $$\int_0^x (x-t+1)g(t)\,\mathrm{d}t = x^4 + x^2,$$ Find $g(x) \, \forall x \geq 0.$ My attempt: First distribute the ...
-3
votes
0answers
17 views

Trapezoidal and Simpson rule? [on hold]

Find the values of the following integral using a)Trapezoidal rule b)SImpson rule $ \int_0^{0.5} [2/(x-4)] dx$
1
vote
3answers
20 views

Indefinite integrals with rati0nal and polynomial functions and Substituion

I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would ...
0
votes
2answers
58 views

How do I solve $\int_{0}^{\infty} \frac{\ln(x)}{1+x^{2}}\,dx$?

If we first split the integral into two: $$\int_{1}^{\infty} \frac{\ln(x)}{1+x^{2}}\,dx$$ and $$\int_{0}^{1} \frac{\ln(x)}{1+x^{2}}\,dx$$ Let $x = 1/u$ and $dx = -1/u^2 du$, then we have: ...
1
vote
2answers
52 views

Deriving a joint cdf from a joint pdf

I see that a similar question was asked last year, but I am still confused. I have $f(x,y) = 2e^{-x-y}$, $ 0 < x < y < \infty $ and need to find the joint CDF. I have a solution that ...
0
votes
2answers
32 views

is it possible to intergrate this function to get x(t) and y(t)?

say you have a function as below; $d^2V(t)/dt = -B^2V(t)$ B is a constant Initial conditions $V_x(0) = V$, $V_y(0) = 0$ I can't see how to integrate to get x(t) and y(t); I ended up with ...
0
votes
1answer
26 views

Calculating surface area

I have the following surface in $$R^3:{(x,y,z),(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2) \ ,\ x,y >=0}.$$ I want to find it's surface area. I've tried using spherical coordinates but calculating the ...
0
votes
0answers
29 views

Does isometry preserve volume on open sets?

Suppose there are two open sets $A,B$. $h$ is an isometry. And the function $h$ maps $A$ to $B$; $h(A)=B$. I need to show that isometry is volume preserving. Any hint would be appreciated! Thanks ...
7
votes
2answers
163 views

how to compute this limit

compute $I=\lim\limits_{n\to+\infty}\sqrt[n]{\int\limits_0^1x^{n+1}(1-x)\cdots(1-x^n)dx}$ attempt: I tried to evaluate the integral $$\begin{align} ...
0
votes
1answer
28 views

integration and convolution

Please can some one help me on the following integration. $$ G(\nu)=\frac{1}{\Delta t}\int_{t_a - \frac{\Delta t}{2}}^{t_a + \frac{\Delta t}{2}} f(t_a -t)e^{-2\pi\nu it}dt $$ where ...
0
votes
3answers
109 views

Solving $\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$

Evaluate $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$$ This is my attempt: $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x} = ...
0
votes
2answers
31 views

derivative and integral as opposite operations

Consider: $$\lim_{y\to\infty} \left( \int_0^y f(t)dt \right)' = \lim_{y\to\infty} f(y)$$ So the integral and the derivative cancel each other, but why is it happened to be that it equals to the ...
1
vote
3answers
62 views

Evaluation of integral $\int_{-\infty}^{+\infty} xe^{-|x|}\,dx$ is not $0$

Given $$f(x)=\frac12e^{-|x|}, -\infty \le x \le +\infty$$ $$\int_{-\infty}^{+\infty} x f(x)\, dx= -\frac12\int_{-\infty}^{+\infty} x (-e^{-|x|})' dx=-\frac12\bigg(-xe^{-|x|} + ...
0
votes
0answers
35 views

indefinite integral problem: help needed

What will be the integral with respect to $t$ of: $$\frac{dA}{dt} = cx(t)y(t),$$ where $c$ is a constant and $x$ and $y$ are functions of time ($t$). Is there any other method besides inegration by ...
1
vote
0answers
21 views

contour integral and limit: What is the condition of the interchange the order?

In real real analysis sense, the interchange between limit and integral is hold when integrand is uniformly converges. $i.e$ \begin{align} \int \lim f = \lim \int f \end{align} Here i want to ...
0
votes
3answers
42 views

Divergence/convergence of an integral

I am told that the following integral converges for $1<n<3$. $$ \int_{-\infty}^{+\infty} (1-e^{ix}) |x|^{-n} dx $$ I am a bit baffled. Anyone with a clue or where to start with this in order to ...
5
votes
5answers
59 views

Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converge.

Prove that $\int_0^1 \frac{\ln x}{x-1} dx$ converges. We cannot apply Abel's/Dirichliet's tests here (For example, Dirichliet's test demands that for $g(x)=\ln x$, $\int_0^1 g(x)dx < \infty$ ...
4
votes
5answers
96 views

Solve $\int_0^\infty \frac{\ln x}{x^2+4} \,\mathrm{d}x$

(Stanford Math Tournament 2012 #8) I tried rewriting the denominator as $4\left(\frac{x}{2}^2 + 1\right)$ and then integrating by parts, but that got me nowhere... I then tried the substitution $x = ...
-1
votes
0answers
64 views
+50

analytic solution of a definite integral

Integrate the following $$\int_0^\infty \alpha\,\beta\, c\, k\, x^s\, x^{c-1} (1+x^c)^{k-1} \left[(1+x^c)^k-1\right]^{-\beta-1} \left[1+\gamma ...
2
votes
2answers
398 views

Splitting an integral

Why is the following equality true? $$ \int_1^{2e} \left| \ln x - 1 \right| dx = \int_1^e(1-\ln x) dx + \int_e^{2e} (\ln x - 1) dx$$
1
vote
1answer
37 views

Show how $\frac{\partial}{\partial x} \left[\int_0^x (x-t)g(t)\,\mathrm{d}t\right] = \int_0^x g(t)\,\mathrm{d}t$

It has something to do with the second part of the Fundamental Theorem of Calculus right? I've always had trouble with this theorem ever since I learned it several years ago :\ Would somebody please ...
0
votes
1answer
40 views

Showing $\sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty$ implies $f_{n}\rightarrow f$ almost everywhere.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of integrable functions and $f$ an integrable function. I have to show that, if $$ \sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty, $$ then ...
1
vote
1answer
17 views

Integration with respect to a measure

I am trying to get an explanation in words, or math, of what the $d\mu$ means in an integration statement. Such as: $$\int f \ d\mu$$ How does the measure change our old "calculus" notion of ...
-3
votes
0answers
37 views

Here is an (QFT, QED related) integration problem in 2+1 D $\vec{k}$ is spacial, $k^{\circ}$ is temporal part. Any suggestions how to proceed? [on hold]

Kindly check it.I don't know what to do with this delta and etc etc $\int \frac {e^{-i\vec{k}.( \vec{x}'-\vec{x})}e^{-ik^{\circ}x_{\circ}}\delta(k^{\circ})}{k^2-\mu^2}d^2kdk^{\circ}$
1
vote
1answer
28 views

Proving an identity using Riemann-Stieltjes Integration?

Prove the following identity using Riemann-Stieltjes Integration: $$\sum_{n=1}^N \frac{1}{n^s} =\frac{1}{N^{s-1}} + s \int_1^N \frac{\lfloor x\rfloor}{x^{s+1}}dx$$ Here's what I have so far: $$ ...