All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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How is Fubini Theorem used here?

Let $\mu$ be a $\sigma$-finite translation invariant measure defined on the Borel subsets of $\mathbb R^d$ and $\lambda$ be the usual Lebesge measure. My question is how the Fubini theorem is used in ...
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18 views

Finding volumes with double integrals

I got some troubles with this problem: A swimming pool is circular with $20\,ft$ diameter. The depth is constant along east-west lines and increases linearly from $4\,ft$ at the south end to $9\,ft$ ...
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gradient of an integral with variable in the upper limit

Is the value of the following computation true? Let $x,y,z\in R^{2n}$ and the continuous vector-valued function $F:\mathbb{R^{2n}}\to\mathbb{R^{2n}}$. Then $\nabla_y \left(\int_{0}^{y}F(x)\cdot ...
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8 views

Limit of a quotient of volumes

Let $f: \Omega \subseteq \mathbb{R}^n \to \mathbb{R}$ be a $C^1$ diffeomorphism defined on a compact subset $\Omega$. I want to see why for any $a \in \Omega$: $\lim_{r \to 0} ...
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Need help verifying two questions on work. One involving Hooke's law, one involving with gas pumping from a tank.

A spring has a natural length of 24 inches. If a force of 120 lbs. is required to compress the spring 5 inches beyond its natural length, then find; a)work done compressing the spring from a length ...
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1answer
19 views

Differentiation under integral sign without DCT

Suppose $f: \Omega \times I \subseteq \mathbb{R}^n \to \mathbb{R}$ is differentiable, where $\Omega$ is measurable and $I$ is an open interval. How do you show that if $\frac{\partial f}{\partial t}$ ...
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an integration question ($y\in\mathbb{R}$, $\beta\in (-1/2,1/2)$ fixed) $f_y(x) = \vert x - y \vert^\beta - \vert x \vert^\beta$ is square-integrable

Fix $y\in\mathbb{R}$, we define a function $$f_y(x) : = \vert x - y \vert^\beta - \vert x \vert^\beta \quad, \quad x\in\mathbb{R} $$ where $\beta$ is a number in $( -1/2, 1/2)$. Show that ...
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1answer
13 views

double integral over an arbitrary triangle

Assume we have an arbitrary triangle ABC in x-y plane and we want to integrate a function $f(x,y)$ over surface of this triangle as shown in fig. 1: We can define another coordination system [x' ...
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3answers
65 views

Integrate $\frac{x}{1+x^2}$

Integrating $\frac{x}{1+x^2}$ becomes $\ln \sqrt{x^2 + 1}$ Why is this? Is there a formula or a fact that makes this so. Integrating this a lot different than integrating something easy like $X^2$.
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78 views

Which integral is larger?

The question: Given $f$ to be a positive, measurable function on $[0,1]$, which is larger, $\displaystyle\int_0^1 f(x)\log f(x)\,dx$ or $\displaystyle\left(\int_0^1f(s)\,ds\right)\left(\int_0^1\log ...
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Normalisation of Monte Carlo overlap

In quantum mechanics you are sometimes required to compute the inner product of two wave functions (square integrable complex functions) $\Psi(x)$ and $\Phi_i(x)$ as $$ \lambda_i=\int ...
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1answer
4 views

Integrability condition for bounded function defined in a block

I need a hint for this question: I need to prove that if $f: A \subseteq \mathbb{R}^n \to \mathbb{R}$ is bounded, where $A$ is an $n$-dimensional box, then $f$ is (Riemann) integrable iff for each ...
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1answer
28 views

Radon-Nikodym: Alternative?

Is it possible to define a support for positive measures so that for positive and negative variation of a real measure one has: $$\mu_+(E)=\int_E\chi_A\mathrm{d}|\mu|$$ ...
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15 views

How to select the integration contour

In the following two figures which describe sets, How many possible integration contour we have for the figure 1 and how many integration contour we have for figure 2.
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1answer
22 views

Trapezoidal rule over interpolation of higher dimensional vectors

According to a wikipedia and mathworld, the trapezoidal rule is: $$ \int_a^b f(x)\,dx \approx h\left[\frac{f(a) + f(b)}{2} \right], $$ where $h = (b-a)$. If you apply this rule to a function ...
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68 views

how to calculuate $\int_0^ \pi \sqrt{1+x^2 \sin^2x}dx$

I was finding arc length of y=sin$x$-$x$cos$x$ $(0 \leq x\leq \pi)$ and I found I've to solve $$\int_0^\pi \sqrt{1 + x^2\sin^2{x}}\, dx $$ but I have no idea about this I tried using ...
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4answers
81 views

Integration without substitution of $\frac{x^2+3}{x^6\left(x^2+1\right)}$

This is a repost of a question i had written incorrectly earlier. How do I integrate this without substitutions ? $$ \frac{x^2+3}{x^6\left(x^2+1\right)} $$ I got: $$ ...
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24 views

Integral Hölder bound

I was wondering if it is possible to find the following bound or if not, find a counterexample of it. Let $f\in C_0^1$ (compactly supported continously differentiable, in particular $\alpha$-Hölder ...
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17 views

Liouville's Extension of Dirichlet Theorem

Can we use Liouville's Extension of Dirichlet Theorem to find triple integral $\int\int\int(u^2+v^2+w^2)\space du\space dv\space dw\space where\space u=0, v=0, w=0\space \&\space u+v+w\leq1$? Or ...
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2answers
49 views

Integration without substitution

How to i integrate this with out substitutions or Partial fraction decomposition ? ($3x^2$+$2$)/[$x^6$($x^2$+1)] I've got to : 2/x^6(x^2+1),but after this i haven't been able to eliminate the 2.
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Finding an optimal path for minimizing an integral.

Let $x,y$ be real numbers. Let the function $f(x,y)$ be real-entire in both $x$ and $y$. Thus $f(x,y)$ is a real-entire Taylor series in the variables $x,y$. How the find a non-intersecting path ...
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4answers
44 views

Definite Integrations problems [on hold]

If $f(x)= x^2 e^{x^2}$ then show that $f'(x)= 2xe^{x^2} + 2x^3 e^{x^2}$ and use this result to evaluate $$\int x^3 e^{x^2} \, dx$$ How can I use the result to evaluate the integral?
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Behaviour of the error function as $z \rightarrow -\infty$?

I'm trying to find the behaviour of the error function, $erf(z)$ as $z \rightarrow -\infty$ $$erf(z) = \frac{2}{\sqrt{\pi}}\int_0^{z} e^{-s^2}\mathrm{ds}$$ I know that we can find the limit of ...
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A question about integral-squared error.

We consider the problem of representing a time function, or signal, $x(t)$ on a $T$-s interval $(t_0, t_0+T)$, as an expansion. Thus we consider a set of time functions $\phi_1 (t), \phi_2(t), ..., ...
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157 views

Simplest way to integrate this trigonometric integral:

$$\int \frac{1}{1+\tan x}dx,$$ A substitution like $t = \tan x, \;dt = (1+t^2)dx$ etc. immediately comes to mind, but I find this method a bit lengthy with the partial fractions. Is there a more ...
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7answers
305 views

Proving area under the integrals.

I have a question that I have been trying to solve that I am curious about. If you have a continuous function $f(x) = \frac1x$. How would you prove that ...
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14 views

Showing equality between integral and shifted Fourier series

Let $f\in E[-\pi,\pi]$ and let $f\sim \frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx +b_n\sin nx$ be the fourier series of f in $[-\pi,\pi]$.show that $$\forall -\pi\le c,x\le \pi \quad ...
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1answer
29 views

What is the domain of the inverse function

$f(x)=(x + 6)/(4x + 3)$ (a) Find the inverse function of $f$. My solution to (a) is $x=(y+6)/(4y+3)$ $(y+6)/(4y+3)=x$ $y+6=4xy+3x$ $y+6-4xy=3x$ $-4xy+y+6=3x$ $y=-(3(x-2))/(4x-1)$ I replaced ...
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3answers
44 views

Evaluating $ \int {e^x \sin (k \pi x) } dx $

I'm trying to integrate $$ I = \int {e^x \sin (k \pi x)} dx. $$ I've used Matlab and Wolfram Alpha, which have both given me the result $$ I = \frac{e^x(\sin (k \pi x) - \cos (k \pi x))}{k^2 \pi^2 +1 ...
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4answers
30 views

Taking partial fractions for integration?

I'm having some trouble with integrals involving partial fractions it seems. Been stuck on this forever. The equation is given below and I have to use partial fractions to solve. ...
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13 views

Hypothesis testing CDF

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
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0answers
13 views

Solve $\int\limits_{\mathbb{R}^n} e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$

How to calculate $$\int_{\mathbb{R}^n}e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$$ where $\langle \cdot, \cdot \rangle$ denotes the canonical inner product in $\mathbb{R}^n$. I'm trying use ...
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0answers
21 views

justification of step in complex integration

What is the justification for the step with the red square next to it, how do we change the integrator like this?
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1answer
26 views

Tough function to integrate

I am having trouble seeing the process to integrate this function (wrt T) $$A(T) = \frac{1}{(CT^4)}\frac{(1-a)K+aT}{a(K-T)}$$ Should I use integration by parts? I do not see how this will work
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1answer
20 views

Solve $\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx $

How to prove that $$\int\limits_{-\infty}^{\infty}e^{-cx^2}\sin(sx)dx = 0,$$ where $c>0$?
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1answer
14 views

volume of solid of rotation: finding r

For a region bounded by: $$y=x+4,\;y=16-x^2;\;around\;y=-5$$ I understand that I will be using the 'washer' method: $$V =\int_a^b\pi r^2h$$ But I'm having a hard time finding $$r^2 \text{ for}= ...
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1answer
19 views

Curvature of curve

$r(t) = (-3sint)i + (-3sint)j + (cost)k$ I got as far as:$$||r'(u)|| = sqrt{(18cos^2u + sin^2u)}$$ But I cannot evaluate $\int_0^t||r'(u)||dt$
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1answer
16 views

Change of Variable involve derivative

Let me just give the 1-D version of my problem. Let $u\in C_c^\infty(R)$ and define $u_r(x):=u(rx)$. Then I am trying to evaluate the integration $\int_R u_r'(x)dx$. Here is my steps: $$\int_R ...
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2answers
65 views

How would you solve $\int \frac{x}{x^2 - 4x + 5} dx$

What is the tip for integrating that integral? I completed the square on the bottom to make it $$\frac{x}{(x-2)^2 + 1}$$ but it doesn't seem helpful. Any tips? Thanks.
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2answers
29 views

Indefinite integral of fraction

I'm working through some indefinite integral exercises. There is one here that I can't seem to figure, and there is no solution in the textbook: $$\int \frac{3}{4x^2+4}dx$$ I'm assuming it has ...
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21 views

Question from integral with using fourier's integral

Please explain me how to compute this integral: $$ \int_0^\infty \dfrac{\cos(\omega x)+\omega \sin(\omega x)}{1+\omega^2}d\omega$$
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1answer
31 views

Trigonometric Integration: Using the half-angle formula?

I'll preface my question by saying this is my first ever post. I've been lurking around and answering a couple logic questions here and there, but since I have an intractable calculus question I ...
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1answer
74 views

intregration without substitution of $x^x \ln x$

How do i integrate this without any substitution, purely algebraically : $$x^x \ln ex$$ I've tried a lot but not have been able to: $$x^x \ln (x + 1) = \ln x^{x^x} + x^x$$ or $e^{x \ln x}\ln ...
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1answer
17 views

How to describe two integration contours as set? [on hold]

Friends I need support to understand how one can describe two integration contours as set? can anyone please explain it with the help of a example?
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1answer
28 views

Finding $\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$ using Plancherel’s theorem

Suppose $G(\mathbb R)\ni f(x),\mathcal{F}[f](\omega)=\frac{1}{1+|w|^3}$ find the value of $$\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$$ I thought using Plancherel’s theorem \begin{align} ...
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1answer
85 views

A cute limit $\lim_{m\to\infty}\left(\left(\sum_{n=1}^{m}\frac{1}{n}\sum_{k=1}^{n-1}\frac{(-1)^k}{k}\right)+\log(2)H_m\right)$

I'm sure that for many of you this is a limit pretty easy to compute, but my concern here is a bit different, and I'd like to know if I can nicely compute it without using special functions. Do you ...
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35 views

What is integration contour and how to discribe it? [on hold]

We knew that an integration contour can be described as a set of points. How one can describe the two integration contours as sets.? Can anyone help me with examples.
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1answer
41 views

Integral using Beta Function and Gamma Function

Interestingly, I seem to have an integral I have posted before, but I want to take a different approach to it. $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} \,dx$ The beta function states, $B(x,y) = ...
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1answer
32 views

2D Fourier transfrom of $1/(x^2-y^2+q)$

How can I calculate the following 2D Fourier integral: $$ \iint \frac{{\rm e}^{{\rm i}(ax+by)}}{x^2-y^2+q} {\rm d}x\,{\rm d}y, $$ where $q$ is a complex number? If there was a "+" sign in the ...
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4answers
88 views

Problem with a solution to the integral $\int_{-\infty}^{+\infty}e^{-x^2}\mathrm{dx}$

I am an undergrad in my first year of college. Today, our mathematics professor solved the integral $\int_{-\infty}^{+\infty}e^{-x^2}\mathrm{dx}$ which he called "One of the most important integrals ...