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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2
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3answers
27 views

Integral of $\cos(\cos x)$ over $[0,2\pi]$

How to compute the following integral? $$\mathcal{J}_2=\int_{0}^{2\pi}\cos(\cos t)\,dt$$ I'm trying to compute this integral, but I have no idea of how to do it, can someone help me?
1
vote
0answers
26 views

Help with functional integral

I'm stuck on how to do a functional integral. The integration I'm trying to do is of this form $\frac{\partial}{\partial B(\tau)} \left[ \exp\left(-B^2(\tau)\right)+\int_{0}^{\tau} ...
0
votes
0answers
11 views

Integrating differential forms over a box

I've only ever seen examples of integrating a differential form over a curve C involving defining a parameterization. I have seen people integrate 1 forms over a box without defining a ...
6
votes
2answers
51 views

Trigonometric functions expressed as definite integrals with Bessel functions

Prove that $$\frac{\sin(x)}{x}=\int_0^\frac{\pi}{2}J_0(x\cos(\theta))\cos(\theta)\,d\theta \tag{a}$$ $$\frac{1-\cos(x)}{x}=\int_0^\frac{\pi}{2}J_1(x\cos(\theta))\,d\theta \tag{b}$$ Hint: ...
2
votes
3answers
94 views

Limit of a Riemann Sum and Integral

I've been trying to solve this problem, but I haven't been able to calculate the exact limit, I've just been able to find some boundaries. I hope you guys can help me with it. Let $f:[0,1] \to ...
4
votes
1answer
66 views

A closed-form of $\frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx$

I am looking for a closed-form of this integral \begin{equation} \frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx \end{equation} I ...
1
vote
1answer
47 views

Find the coefficients in quadrature formula on $[0,1]$ with the nodes at $1/4$, $1/2$, $3/4$

In my worksheet I was given a question about numerical integration that says: Find the formula for $\int_{0}^{1}f(x)dx=A_{0}f(\frac{1}{4})+A_1f(\frac{1}{2})+A_2f(\frac{3}{4})$ I suppose the goal ...
1
vote
0answers
32 views

How can I calculate the force that is applied on a tube by an another tube?

Let's say there is two tubes (cylinders with no tops or bottoms) with charges $q_1$ and $q_2$, radii $b_1$ and $b_2$, lengths $\ell_1$ and $\ell_2$. These tubes are located along the axis of each ...
2
votes
0answers
9 views

Calculating $\text{D}g$ of $g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t$

Let $g:(1,\infty)^2\to\mathbb{R}$ be given by $$g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t.$$ How can I calculate $\text{D}g$ using parameter-dependent integrals?
2
votes
0answers
33 views

What is an example of a function that is measurable but not strongly measurable?

Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$. $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of ...
6
votes
3answers
136 views

Find $\int_0^\pi \sin(x)\,dx$ explicitly

A book asks me to prove that: $$\int_0^{\pi}\sin(x)\,dx = 2$$ Using the identity: $$\sin\left(\frac{\pi}{n}\right) + \sin\left(\frac{2\pi}{n}\right) + \cdots + \sin\left(\frac{n\pi}{n}\right) = ...
4
votes
0answers
66 views

is there closed form for $\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})dx$

Is there closed form for $$I(a)=\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})dx $$where is $a\in (-1,3)$ I've tried with $\tan x=u$ and I got the result of sum in term of ...
1
vote
0answers
18 views

Will Gauss quadrature numerical integration work with a variable dx

The question kind of says it all, but I'm reading about Gauss quadrature from here: http://www.damtp.cam.ac.uk/lab/people/sd/lectures/nummeth98/integration.htm which gives an equation of this form: ...
2
votes
3answers
101 views

What is $\operatorname{Ei}(x)$?

I was trying to solve $$\int\frac{e^x - e^{-x}}{x}\,dx$$ But I have no idea how to do it and the calculator said to use a common integral that I don't know what it means.
12
votes
2answers
179 views

Integral $\int_0^{\Large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx$

I am wondering if anyone would know how to evaluate this integral: $$\int_{0}^{\Large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx.$$ I've tried, unsuccessfully, the change ...
2
votes
1answer
43 views

Positive Linear Transformations: What good for?

Positivity is a concept appearing quite frequently in the study of algebras and its related spectral theory. Positive elements naturally give rise to an ordering and therefore allows to construct ...
0
votes
1answer
33 views

Integrating probabilities

My following problem is of general nature, here is an example to illustrate it. For example let $\left(\xi_i\right)_{i \geq 1}$ be independent and identically Exp(1) distributed random variables. We ...
1
vote
2answers
48 views

Find the volume below $\sqrt{x}+\sqrt{y}+\sqrt{z}=1$ in the first quadrant

I understand that we have to use transformation $$x = u^2, y = v^2, z = w^2$$ but I cannot figure out the limits. I just need a rough sketch of how to approach this. Could anyone give me some ideas?
2
votes
1answer
81 views

Antiderivative of $\exp(x^2)$ [duplicate]

Can you please provide a step by step solution for next integral. I don''t have any idea of hoe this can be solved $\displaystyle\int e^{x^2}\,dx$.
13
votes
2answers
86 views

Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
4
votes
2answers
62 views

Integral of the Square of the Elliptic Integral

Someone must know a good technique for $$ \int E^{2}(x)dx $$ Where $E$ is the complete elliptic integral of the second kind: $$ ...
2
votes
0answers
41 views

Simple Integral Involving the Square of the Elliptic Integral

I have, $$ \int uE^{2}\left(u\right)du $$ where $E$ is the complete elliptic integral of the second kind: $$ E\left(k\right)=\int_{0}^{\frac{\pi}{2}}d\theta\sqrt{1-k^{2}\sin^{2}\left(\theta\right)} ...
6
votes
0answers
89 views

Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
2
votes
6answers
133 views

Find $\int_{-1}^1 (8x^3 + 14x^2 + 6x + 3)dx$. [on hold]

Does anyone know the answer to this integration? $$\int_{-1}^1 (8x^3 + 14x^2 + 6x + 3)dx$$
1
vote
0answers
35 views

Finding the mean value of y

I don't understand how to obtain the limits for the $t$-values considering that they gave us the $x$-values in the first part of the equation. I've considered substituting the $x$-values into the ...
0
votes
1answer
24 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
0
votes
1answer
30 views

Area of a triangle - straight lines

Question: What is the area of the triangle formed by the line $x + y = 3$ and angle bisectors of the pair of straight lines $x^2 - y^2 + 2y = 1$. Well I really have no idea how to even start the ...
1
vote
1answer
44 views

Integrating a Ratio of Elliptic Integrals

Can anyone help evaluate $$\int dx\frac{\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\left(\theta\right)}}}{x\int_{0}^{\frac{\pi}{2}}d\theta\sqrt{1-k^{2}\sin^{2}\left(\theta\right)}}$$ ...
-4
votes
1answer
42 views

Let $f(t)\in \mathcal C'[-1,1]$. Evaluate $\lim_{n\to \infty}\frac 1 n \sum_{k=1}^nf'\left(\frac k {3n}\right)$. [on hold]

Let $f(t)\in \mathcal C^1[-1,1]$. Evaluate $$\lim_{n\to \infty}\frac 1 n \sum_{k=1}^nf'\left(\frac k {3n}\right)$$
1
vote
4answers
58 views

If $f(t)\in \mathcal{C}[-1,1]$ then evaluate $\lim_{h\to\infty} \frac{1}{h}\int_{-h}^hf(t)dt$.

If $f(t)\in \mathcal{C}[-1,1]$ then evaluate $$\lim_{h\to 0} \frac{1}{h}\int_{-h}^hf(t)dt$$ I have just used fundamental theorem of integral calculus. However, I could not estimate this...that ...
0
votes
2answers
143 views

What is $\int x! $ $ dx$?

What is $\int x! $ $ dx$. $f(x)=x! $ looks something like this. Do we have any formula for finding this indefinite integral.
6
votes
2answers
105 views

Integral $ \int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$

Please help evaluating this integral $$ \large\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$$ Mathematica could not evaluate it in a closed form. Numerically it is about ...
2
votes
2answers
75 views

How to determine $\int e^{2x} \sqrt{e^x+1}dx$?

Determine $\int e^{2x} \sqrt{e^x+1}dx$ Is there a multiplication rule for integration or something?
0
votes
3answers
35 views

Calculus (what is y when x is?)

Given $y>0$ and $$dy/dx = (3x^2+4x)/y$$ If the point $(1,rad10)$ is on the graph relating x and y, then what is $y$ when $x=0$? I'm not sure whether or not to integrate, or just plug in the ...
-2
votes
4answers
60 views

Calculus (limits)

Compute $$\lim_{t\to0}\frac{\tan\left(\dfrac {1}4\pi + t\right) - \tan\left(\dfrac{1}4\pi\right)}t$$ Alright, so I'm taking the derivative first. Is there an easier way to take the derivative of ...
0
votes
1answer
58 views

Calculus (advanced integration)

Compute $\int (5^x+2e^{5 \ln x})dx$ The $5\ln x$ part confuses me. So far I have $5^x/\ln 5\:\:$
1
vote
0answers
91 views

Problem with trigonometric substitution proof

I'm sad, I can't get it. I know perfectly how to integrate using the mechanical process described in the books, but I want to understand the proof of it. My book (Stewart) says: In general we can ...
5
votes
3answers
203 views

Calculus (Integration)

Is there a simple way to integrate $\displaystyle\int\limits_{0}^{1/2}\dfrac{4}{1+4t^2}\,dt$ I have no idea how to go about doing this. The fraction in the denominator is what's confusing me. I tried ...
1
vote
2answers
39 views

Integration by parts

Integrate using integration by parts: $F(y) = (y+1)e^{-y}$ Find: Evaluate the $\int_{a=0}^{b=\infty}F(y)\;dy$ using integration by parts. Thus far, I've distributed the $e^y$ term and split ...
2
votes
1answer
48 views

Generalized Logarithmic Integral - reference request

This page at I&S forum defines the Generalized Logarithmic Integral as $$L\left[ \begin{matrix} a,b,c \\ d,e,f \end{matrix};z\right] =\int_0^z \frac{\log^a x \log^b(1-x)\log^c(1+x)}{x^d (1-x)^e ...
1
vote
1answer
54 views

Derive the formula for the sum of the first $n$ squares using derivatives and integrals

I wanted to prove the formula for sum of squares without using induction and thought using derivatives would be the easiest approach ...
1
vote
0answers
24 views

Counterexample for necessary condition of integrability

Can you give me an example of a non-negative function on $[0,1]$ that is NOT integrable, but $\lim_{t \to \infty} t \mu\{x : |f(x)| \geq t \} =0$?
2
votes
1answer
53 views

Double Integral of a piecewise function

If $F(x,y)$ is defined as $F(x,y) = x+y$ when $0 < x + y < 1$ and $0$ elsewhere, then find $$\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} F(x, y) \,dx \,dy$$. Math note: I've ...
1
vote
1answer
21 views

Expected length of a random vector

I meet a basic definition about the expected length of a random vector when reading a paper: What is "expected length" How to roughly derive both equations (yellow part) (Is that Gamma ...
2
votes
1answer
54 views

Indefinite integration of $1/\sqrt{3-5x-2x^2}$

Cannot make it out. $$\int \frac{dx}{(3-5x-2x^2)^{1/2}} $$ Is the problem correct, or does it have errors? I have a doubt.
2
votes
0answers
22 views

Normal Vector Affecting The Divergence Theorem

$\newcommand{\Div}{\operatorname{Div}}$I'm going to use an example to explain what I'm trying to ask. Let $T =\{(x,y,z): x^2+y^2=z^2, 0\leq z\leq3\}$, I'm asked to calculate $\iint_T ...
1
vote
2answers
56 views

Finding the integral of $(x^2+4x)/\sqrt{x^2+2x+2}$

Can somebody explain me how to calculate this integral? $$\int \frac{\left(x^2+4x\right)}{\sqrt{x^2+2x+2}}dx$$
2
votes
0answers
64 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration$$ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$$ I am not sure as to how to work with the branch cuts of both $\ln{x}$ and $\sqrt{1-x^2}$ ...
4
votes
1answer
73 views

Integral of $1/(1+x \tan(x))^2$

How would you solve the following integral? $$\int \frac{1}{(1+x\tan(x))^2} dx$$ Any help would be appreciated.
0
votes
2answers
58 views

A identity relating a infinite series and a definite integral [duplicate]

Prove that, $$ \sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!