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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3
votes
0answers
146 views

Evaluate $\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$

We have the following integral: $$\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$$ where $P_3(t)$ is a third-degree polynomial with all coefficients different from zero. Is it an elliptic integral? ...
2
votes
1answer
43 views

If a function is enclosed by lower and upper sums, does its limit w.r.t. partitions equals the integral

Let $a < b$ and denote by $\mathcal P[a,b]$ the set of all finite partitions of the compact interval $[a,b]$, i.e. all sets of the form $P = \{ a = x_0 < x_1 < \ldots < x_n = b \}$. ...
11
votes
3answers
212 views

About the integral $\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx$

I came across the following Integral and have been completely stumped by it. $$\large\int_{0}^{1}\dfrac{\log(x)\log^2(1+x)}{x}dx$$ I'm extremely sorry, but the only thing I noticed was that the ...
4
votes
2answers
85 views

Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$.

Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is ...
1
vote
2answers
80 views

Calculate $\int\int{\sqrt{1-x^2-y^2}}dxdy$ the domain is $x^2+y^2 \le x$

The question: Calculate $\int\int{\sqrt{1-x^2-y^2}}dxdy$ the domain is $x^2+y^2 \le x$ my solution: the correct answer offered by my teacher : I can't figure out why I am wrong. I wonder which ...
6
votes
0answers
134 views

Fourier sine transform of $\frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert$

Show that $$ \int_0^{\infty} kF(k)\sin(ka)\,dk = \frac{\pi}{2}aG(a) $$ where $$ F(x) = \frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert $$ and $$ G(x) = \frac{\sin x-x\cos x}{x^4} $$ EDIT: ...
5
votes
1answer
126 views

Integration by parts for general measure?

Let $\mu$ be a general measure, suppose $f,g$ has compact support on $\mathbb{R}$, when does the integration by parts formula hold $$\int f'g d\mu = - \int g'fd\mu?$$ I know in general this is false, ...
6
votes
6answers
287 views

Compute definite integral

Question: Compute $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2}dx.$$ Attempt: I've tried various substitutions with no success. Looked for a possible contour integration by converting this into a rational ...
1
vote
4answers
129 views

Solving $\int \frac{dx}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$

This was in an old exam in a physics for mathematicians class. I haven't had to deal with these kind of integrals for a while and can't think of a decent substitution. I asked my teacher about it and ...
1
vote
0answers
141 views

Determine the minimum of $\int_0^\infty\left|x^3+ax^2+bx+c\right|e^{-x}dx$

This question appeared on a graduate preliminary exam in real analysis. Determine $$\min_{a,b,c\in\mathbb{R}} \int_0^\infty\left|x^3+ax^2+bx+c\right|e^{-x}dx.$$
2
votes
1answer
36 views

What exactly is “integrated form”?

I am reading on population growth and I see $\Delta N_t = (b - d)N_t \, \Delta t = mN_t \, \Delta t$ , where $m = b - d$. As $\Delta t \to 0$, this becomes $\dfrac{dN_t}{dt} = ...
0
votes
0answers
32 views

Relation between $\lim_{a \to 0}\int_a^T u(t)$ and the Lebesgue integral $\int_0^T u(t)$

Let $u\colon (0,T] \to \mathbb{R}$ be function with $u \geq 0$ everywhere and $u$ is continuous on $[a,T]$ for every $a > 0$. Suppose that the limit $$\lim_{a \to 0}\int_a^T u(t) \;dt ...
1
vote
1answer
74 views

What is $\int\sinh(x)^pdx$?

What is $$\int\sinh(x)^pdx$$, where $0<p<1$?. I tried using Mathematica, but it came up with some Hypergeometric2F1 function. Is there a simpler answer in this integral?
7
votes
2answers
91 views

Lebesgue integrable function over $(0,1)$ vs $[0,1]$

Up till now, I thought saying $u \in L^2([0,1])$ is the same as saying $u \in L^2((0,1))$, because I see people emphasizing "$u$ is Lebesgue integrable over $[0,1)$". I thought the whole point of the ...
0
votes
1answer
96 views

How do I integrate $\frac{\sin x+\cos x}{\sin^4 x+\cos^4 x}$ [duplicate]

How do I integrate $$\frac{\sin x+\cos x}{\sin^4 x+\cos^4 x}$$ ? Tried different ways including the tangent half-angle substitution (which seems to be disastrous).
0
votes
0answers
44 views

Can I have variables extreme of integration?

Suppose you have a function $v(t)$ that you want to find. The condition is that it's integral is some fixed quantity. The integral is done between $0$ and $u(t)$, where $u(t)$ is an increasing ...
1
vote
1answer
40 views

When can I take $\lim_{a \to 0}\int_a^T u$?

Suppose I have a function $u:(0,T) \to \mathbb{R}$ which is integrable over $[a,T]$ for every $a > 0$, and I have the results $$\int_a^T u = U(T)-U(a)$$ for such $a$. When am I allowed to conclude ...
0
votes
0answers
38 views

Approximation Lemma for Riemann-integrable functions

In the following let $f : [a,b] \to \mathbb R$ be bounded functions. For a regulated function, the integral could be written as the limit $$ \int_a^b f(x) dx = \lim_{n\to \infty} \int_a^b ...
2
votes
1answer
99 views

Arclength of intersection between 2 perpendicular cylinders.

hi have 2 perpendicular cylinders that intersect (I read the resulting curve is called the Steinmetz curve). $x^2+y^2=R_1^2$ and $y^2+z^2=R_2^2$, with $R_2\lt R_1$ and want to parametrize the length ...
0
votes
2answers
96 views

Bounded function on compact interval that is not Lebesgue integrable

Is there an example of a bounded function $f : [a,b] \to \mathbb R$ which is not Lebesgue integrable?
0
votes
0answers
35 views

Composite trapezoid rule and trigonometric functions

I am trying to solve the problem talked about in: Trapezoid rule over trigonometric polynomials Show that the composite trapezoid rule over an equidistant partitioning with interval size ...
1
vote
0answers
79 views

Make this integral zero

Consider the integral $$\int_0^\infty x^nf(x)\,\mathrm{d}x$$ from this answer. The integral is zero for the following $n$ and $f(x)$: $n=4k$, $f(x)=e^{-x}x^{-1}\sin(x)$ $n=4k+1$, ...
0
votes
1answer
60 views

Evaluate $ \int_{\varepsilon}^1 \sin\left( \frac{1}{x} \right) dx$

Let $0 < \varepsilon < 1$, how to solve the integral: $$ \int_{\varepsilon}^1 \sin\left( \frac{1}{x} \right) dx $$
2
votes
2answers
56 views

Prove that a classical solution of $-\langle\nabla,A\nabla u\rangle=f$ is also a weak one

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable and $A(x)$ be symmetric, for all $x\in\Omega$ $u\in C^2(\Omega)$ with $A\nabla ...
0
votes
0answers
178 views

Riemann-integrable iff pointwise limit of step functions

Is the following true? Let $f : [a,b] \to \mathbb R$ be a bounded function: The function $f$ is Riemann-integrable if and only if there exists a sequence of step functions $\varphi_n$ converging ...
-3
votes
2answers
78 views

rockets in the corners of a square [duplicate]

There is one rocket in each corner of a square. At some point they start moving towards the rocket in the neighbouring corner in (say) clockwise direction. Their subsequent motion is such that they ...
1
vote
2answers
101 views

if integral $f(x)\cdot g(x)=0$ mean that $f(x)=0$?

The question: If $f(x)$ is a continuous function, such that for every continuous function $g(x)$ defined over $[a,b]$ $$\int_a^b f(x)\cdot g(x)\,dx =0$$ does it mean that $f\equiv 0$? The ...
0
votes
1answer
38 views

Do equal rational integrands imply equal integrals, save for a constant?

Specifically, when integrating $\frac{1}{ax+b}$ we get $\frac{1}{a}\ln|ax+b|$. However, if we multiply the integrand by say $c/c = 1$, then the integral computes to $(1/a)\ln|c(ax+b)|$. Can ...
1
vote
2answers
67 views

Find the sign of the following integral: $\int_{0}^{2\pi}{\sin x\over x}$.

Am I required to know if it is positive or negative? This expression is not really integrable, not within the boundaries of the course at least. I tried comparing it to other integrals but I get zero ...
1
vote
1answer
22 views

Quadrature on segment

Is there a quadrature formula on the segment [0,1], such that on [0,1/2] the points and weights are symetric with respect to 1/4, on [1/2,1] they are symetric with respect to 3/4, and such that the ...
1
vote
2answers
152 views

Solving a given complex integral

I am trying to solve a problem that involves solving the integral $$\int\frac{1}{\sqrt{y^2 + a^2}} \left(\frac{\sqrt{y^2 + a^2}}{k} - 1\right)^pdy$$ Where $$p=1-\frac{1}{1+n}, n>1$$$, $n$ is an ...
2
votes
1answer
158 views

Integrating over a specific vector field

I am trying to show that the solution of the following integral is as follows: Define the stopping time: $C(a) = \inf(u \ge 0 : H(\pi(0) |\mu)-H(\pi(u) | \mu) > a)$ Where ...
0
votes
1answer
13 views

How to use Jacobi determinant in this simple transformation?

I have two variables:$\vec{x}_1$ and $\vec{x}_2$, then I introduce $\vec{\xi}_1 = \vec{x}_1+\vec{x}_2$ and $\vec{\xi}_2 = \vec{x}_1-\vec{x}_2$. How does the volume element: $\int d^3 \vec{x}_1 d^3 ...
1
vote
1answer
40 views

Integration with Polar Coordinates

I want to integrate this integral with polar coordinates: $\int \sin x \ dA$ on the region bounded by $ y=x, y=10-x^2, x=0$. So far I've got that $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} ...
1
vote
2answers
64 views

Solving recurrence using analogy with continuous $x_{n+1} = \frac{r^2}{2d - x_n}$

What's up lovely friends, I'm facing a physics problem and felt on a recurrence that one does not see everyday. This one: $x_{n+1} = \frac{r^2}{2d - x_n}$ or $f(n+1) = \frac{a}{b-f(n)}$ if you will ...
1
vote
1answer
52 views

About integration by substitution

I know how the method goes: we want to find $\int{f(g(x))g'(x)dx}$, which by the reverse chain rule equals $\int{f(u)du}$. My (maybe stupid) question comes from the integrals with the form ...
11
votes
4answers
178 views

Integrate $\frac{1}{(1+x^2)(1+x^c)}$ from $0$ to $\infty$ for any $c$.

The question is to evaluate $$ \int_0^\infty \frac{dx}{(1+x^2)(1+x^c)} $$ for arbitrary $c\geq0$. Here are my attempts: (The methods behave somewhat differently for $c=0$ but that case is trivial so ...
1
vote
0answers
44 views

How to solve for a Phase Function Cumulative Distribution function (CDF) calculation give a pdf …

I am attempting to solve for the CDF (more specifically the inverse CDF, but that is easy once I have the CDF) - Cumulative Distribution function given a Probability Distribution Function (pdf) and g ...
1
vote
1answer
65 views

Find the derivative of an integral with respect to the upper limit

Let $f(t,y): \mathbb R^2 \to \mathbb R$ be a continuous function of two variables and let $\phi:\mathbb R\to\mathbb R$ be a continuous function of one variable. Fix $a\in\mathbb R$. Compute ...
0
votes
0answers
27 views

Finding area by integration, increasing inaccuracies with complex functions?

I am looking for an explanation as to why the method of integration to find the area of function using limits provides a greater % difference between other methods (In this example Simpsons) with ...
0
votes
0answers
35 views

Error in Gauss-Legendre quadrature

I've tried Googleing this, but so far I haven't succeeded. Can someone point to me a webpage or book in which I can find the error estimate (detailed, not just the final formula) of the Gauss-Legendre ...
0
votes
1answer
67 views

Infinite sum with an improper integral

Trying to find the variance of the logistic distribution I have come to this expression. I tried to solve the integral separately, but this procedure is not working. Can someone guide me on how to ...
2
votes
3answers
72 views

How do I integrate this exponential + Bessel function term?

I would like to integrate this in my research: $\int_0^\infty s e^{i bs^2}J_0(a s)$, where a and b are both real and greater than zero. Integration by parts seems like the obvious first step, but ...
4
votes
3answers
76 views

Help with a Series (Edited)

The original problem was: $$\sum_{k=0}^\infty\dfrac{k}{6k^3+13k^2+9k+2}$$ Using Partial Fractions, I resolved this into ...
9
votes
2answers
131 views

Integral of binomial coefficients

Let the integral in question be given by \begin{align} f_{n}(x) = \int_{1}^{x} \binom{t-1}{n} \, dt. \end{align} The integral can also be seen in the form \begin{align} f_{n}(x) = \frac{1}{n!} \, ...
1
vote
0answers
35 views

Prove that an integral is zero (from Gardiner's Handbook of stochastic methods)

I have troubles in one proof of the book Handbook of stochastic methods by Gardiner. In the paragraph 3.7.3 he writes this integral $\sum_i\int d\vec x \frac{\partial}{\partial ...
6
votes
4answers
102 views

$\int \limits_0^{\infty} x^2 \exp(-2x^2) dx$

How to evaluate this integral? $$\int \limits_0^{\infty} x^2 \exp(-2x^2) dx$$ I found similar problem, but don't know how to apply them here. What do I have to substitute?
3
votes
3answers
99 views

integral of $\sin(\ln(x))\,dx$

I tried to calculate this integral: $$\int \sin(\ln(x))\,dx$$ but it seems my result is wrong. My calculation is: $$\int \sin(\ln(x)) \,dx = \left| \begin{array}{c} u=\ln x \\ du=(1/x)dx\\ ...
0
votes
1answer
51 views

Why does a line integral not depend on the parametrization you use?

I have a question about my calculus course: Why is it true that a line integral over a certain functiondoes not depend on the parametrization you use?. For example, take a function $f(x,y,z)$ of 3 ...
9
votes
3answers
153 views

difficult problem in riemman integrals

Could anyone help me with the following problem? Because i have stuck. problem Let $f:[a,b]\rightarrow [0,\infty)$ be continuous and not the zero function. Prove that $$\lim_{n\to \infty} ...