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 ...

learn more… | top users | synonyms (2)

0
votes
0answers
4 views

Spectral Measure: Support

The support of a spectral measure is defined by: $$\mathrm{supp}E:=\bigcap_{C:E(C)=1}C$$ where $C$ are closed subsets (see german wiki). So by definition it is closed. However I'm wondering wether it ...
0
votes
0answers
12 views

Calculating Area relative to the y-axis

I was asked to calculate the area of the region bounded by the following graph: $$ y = x^2+4x ; y=0$$ I substituted $y$ in order to get $x = 0$ 0r $x=4$. Now I would like a little bit of help to get ...
1
vote
1answer
24 views

Spectral Measure Integration: Product

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. Define the integral of simple functions by: $$\int_\Omega ...
0
votes
1answer
44 views

Area enclosed by curves

Given the curves $y=x^2$ and $y=\frac{1}{2}(x+x^4)$. What is the area enclosed by them ? I can't find the points of intersection of the curves.
13
votes
0answers
54 views

Computing $\lim\limits_{n\to\infty} \Big(\sum\limits_{i = 1}^n \sum\limits_{j = 1}^n \frac1{i^2+j^2}-\frac{\pi}{2} \log(n)\Big)$.

In the chatroom we discussed about the asymptotic of $\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}$, and if we think of the inverse tangent integral, it's easy to see that ...
2
votes
1answer
21 views

Volume generated by lemniscate revolving about a tangent at the pole.

The lemniscates $r^2 = a^2\cos2\theta$ revolves about a tangent at the pole. What is the volume generated by it ? Please explain in detail. I found a couple of answers on finding surface areas, ...
6
votes
7answers
96 views

How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$

How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$) $$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$ Integration by parts doesn't seem to make ...
2
votes
3answers
278 views

Evaluating the integral $ \int{\frac{x}{\sqrt{2x^2 + 3}}}dx $

I am trying to integrate the following: $$ \int{\frac{x}{\sqrt{2x^2 + 3}}}dx $$ It seems to me to be a trig substitution; however, I couldn't seem to get it into one of the three forms, i.e., ...
1
vote
1answer
52 views

Bochner: Lebesgue Obsolete?

Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion ...
0
votes
3answers
33 views

Triple Integral in Spherical Coordinates.

$\newcommand{\de}{\operatorname{d}}$A little stuck on this one. $$\iiint_V ye^{-(x^2+y^2+z^2)^2}\de V$$ Use Spherical Coordinates to evaluate where V is the solid that lies between y=0 and the ...
1
vote
2answers
42 views

Elias Stein : Real Analysis

I cannot understand why this particular line in the text is true: " Moreover, there are $O(k^{d-1})$ cubes in $\cal{Q}\ '$ " For the text see ...
1
vote
1answer
18 views

Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
0
votes
3answers
29 views

Solving using integrating factor [on hold]

Q) Solve $y' = 2x + y$ using the integrating factor. Can anyone guide me with steps here? Help appreciated. Thanks.
3
votes
0answers
27 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ This could be used to evaluate partial sums using knowledge of the ...
3
votes
1answer
39 views

Calc 2: Integration by Parts w/ trig identities

$$\int e^{3\theta}\sec^4(e^{3\theta})\tan^{11}(e^{3\theta})d\theta$$ I just want to make sure that I'm doing this correctly so that I can understand the material. I would also appreciate any tips or ...
2
votes
3answers
78 views

Integral $\int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x}$

Integrate: $$ \int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x} $$
0
votes
0answers
30 views

Evaluating integral involving product of cosine inverse

I am trying to evaluate the below mentioned integral which involves product of two cosine inverses and two variables $x$ and $y$. I need to evaluate the integral or get an approximate value of this ...
1
vote
0answers
12 views

Generalized change of variables in integral

When I read the following (http://www.math.helsinki.fi/~analysis/GraduateSchool/maly/gs.pdf ), it is hard to understand it. In particular, what does it mean by the last equation? Why does it make ...
3
votes
5answers
214 views

Why consider square-integrable functions?

Why are $L^2$ functions important? From reading around I have three hypotheses: they show up in QM (but, why?) they form an inner product space (but, is that a "tight bound" or is the class easily ...
0
votes
0answers
53 views

Is there a formal proof of this basic integral property?

This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth ...
1
vote
0answers
52 views

Prove there exist a $p$ so that the inequality holds

I am stuck with the following problem. Given the Gaussian mixture distribution $f(\cdot)$ $$ f(x) = ...
0
votes
0answers
48 views

Solution to the Integral

I am trying to solve a pdf which contains the following integral. The integral would involve the inverse of cosine function. Can anybody provide me the method how to solve the below mentioned ...
1
vote
3answers
46 views

Integration by parts: $\int e^{-\theta}\cos7\theta \;d\theta$

$$\int e^{-\theta}\cos7\theta \;d\theta$$ I started off by using $u=\cos 7\theta$ and$ \;dv=e^{-\theta}d\theta$, however, this just led me in a circle. I am now at: $$u=e^{-\theta},\;dv=\cos 7\theta ...
0
votes
2answers
34 views

Calculus 2: Strategy for Integration, Integral of e^(x+e^x)dx

How would you find $\int e^{x+e^x}dx$? I know I need to use $u$-substitution but I tried changing what I use for $u$ but I still could not get the right answer. If someone could push me in the ...
2
votes
1answer
42 views

A proof involving nested integrals and induction [duplicate]

Prove that $$\int_0^x dx_1 \int_0^{x_1}dx_2 \cdots \int_0^{x_{n-1}}f(x_n) \, dx_n =\frac{1}{(n-1)!}\int_0^x (x-t)^{n-1}f(t) \, dt$$ I'm trying induction over $n$. The case $n=1$ is trivial. When ...
0
votes
1answer
34 views

Integral Test for convergence of a series

"Consider the series given by $$\sum_{n=2}^{+\infty}\frac{1}{n\ln n(\ln(\ln n))^{\alpha}}$$ for $\alpha>1$. Use the Integral Testo to conclude if the series is convergent or not." I tried to make ...
1
vote
1answer
26 views

Lebesgue Dominated Convergence: Alternative Proof?

Is there an alternative proof of Lebesgue's dominated convergence theorem relying on positive functions only? The point is I'd like to prove that for positive functions: $$\int ...
0
votes
1answer
42 views

How to find $F(x) = \int_x^{x^2} (2+\sqrt t )\, dt$ ?

I have this problem: $$ F(x) = \int_x^{x^2} (2+\sqrt t )\, dt $$ I have to solve the integral. I got $2x^2+\frac{2x^3}{3}-2x-\frac{2x^{3/2}}{3}$ However, I don't think that it correct.
2
votes
3answers
236 views

Why we use dummy variables in integral?

I want to know why we use dummy variables in integral? thanks so much.
3
votes
2answers
133 views

Why does integration of acceleration data create a slope?

I created a 100hz sine wave in code. When I graph the waveform I get this: When I do an integration on this pure sine wave to get a velocity waveform I get: Is this normal? I do not have a ...
0
votes
3answers
61 views

Evaluate the integral $\int_0^{1/4}\frac{x-1}{\sqrt{x}-1}\mathrm dx$

so I have this Integral I have to solve without a calculator. $$\int_0^{1/4}\dfrac{x-1}{\sqrt{x}-1}\mathrm dx.$$ How would I go about finding the antiderivative of that fraction?
-2
votes
1answer
30 views

Evaluating an integral with unspecified functions $f,g$, given other integrals with these functions

Suppose that $$\int_6^8(3f(x)-x)\,\mathrm dx=6$$ and $$\int_8^6(2x+4g(x))\,\mathrm dx=-8$$ Evaluate $$\int_8^6 (f(x)-5g(x))\,\mathrm dx$$ I have a problem. So, this one question asks me ...
1
vote
0answers
15 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
1
vote
0answers
44 views

Prove the given two integrals are not equal

I am stuck with following problem: Prove the following two integrals are not equal: $$ \int_{-\infty}^{\infty} p(y-c)\log \big(p(y-c)+p(y+c)\big)dy \neq \int_{-\infty}^{\infty} p(y+c)\log ...
2
votes
1answer
49 views

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$ This looks like an interesting problem with an elegant ...
0
votes
1answer
28 views

Proving an integration with a modified Bessel function and an exponential

I am trying to prove the following identity: where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of the ...
0
votes
0answers
54 views

How can I evaluate this integral?? [duplicate]

integral $\int_{0}^{\infty} \frac{cosx}{x^2+1} dx$? I got the answer is $\frac{\pi}{2e}$ by using Wolfram. But can't do it by myself... need some help
0
votes
0answers
28 views

Bochner vs. Lebesgue

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
0
votes
4answers
66 views

What is the most efficient way to integrate $(x-3)\sqrt{x^2+3x-18}$?

I can do the problem, but it is becoming so big,that I do not feel to do it anymore. Can anyone give the shortest method for this problem? $$\int (x-3)\sqrt{x^2+3x-18}\,dx $$
0
votes
1answer
34 views

how to remove modulus signs after integrating

$$ \frac{dy}{dt} + k\frac{t^2 -3t + 2}{t+1}y = 0,\ \ \ \ \ \ \ y(t_0=0)=A>0\\ -\int \frac{k}{y} dy = \int (t-4 + \frac{6}{t+1}) dx $$ After integrating the above how do you express $y$ in terms ...
4
votes
1answer
54 views

Integrate $\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$

Integrate $$I=\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$$ Let $$\begin{align}u^2=\frac{\sin(x-a)}{\sin(x+a)}\implies ...
3
votes
2answers
30 views

Existence and uniqueness of weights for the rule $\int_a^b f(x) \ = \ \sum_{0 \leq k \leq n} w_k f(x_k)$

I want to establish this statement: If $a<b$ and $\{x_0,x_1, \cdots x_n\} \subset \mathbb{R}$ distinct, then there is one and only one set of weights $\{w_0, \cdots w_n \} $ such that $\int_a^b ...
1
vote
3answers
36 views

integrate $\int e^{-iwt}dt$

I have this integral: $$ \int e^{-iwt}dt$$ I know that $\int e^{kx}=\frac{e^{kx}}{k}$ so therefore the $ \int e^{-iwt}dt$ would be $\frac{e^{-iwt}}{-iw}$ but Wolfram Alpha says that it is $\int ...
7
votes
0answers
99 views

Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$$ My first thought was to use beta function, but it's hard to get ...
0
votes
2answers
45 views

Can anybody prove why integral of f*f from 0 to 1 not 0? [on hold]

If I have a function f, which can be all real polynomials, Why integral of f * f on [0,1] is not equal to 0 ? I know intuitively, but I need to see the proof
0
votes
2answers
33 views

Integrating this improper integral to test for convergence?

I'm trying to integrate this: $$\int^\infty_0 \frac{8}{\sqrt{e^{x}-x}} \,dx$$ And use the Direct Comparison Test to find out whether it diverges or converges. I looked at a similar problem: and I ...
0
votes
1answer
73 views

Finding total work by integration

The following tank is completely filled with water. Find the total amount of work done in pumping water out of the outlet. Note that the density of water is 1000 kg/m$^3$ I feel like I am ...
0
votes
0answers
8 views

Integral formulation for LDE

I am trying to put the system in a integral formulation. All goes well for the first integration as I obtain What I don't know is how to perform the second integration in this last term. My ...
1
vote
2answers
60 views

Indefenite Integral requiring substitution

Can someone please help me find a useful substitution for the following integral: $$\int \frac{1}{\sqrt{x}(1+\sqrt{x})^2}dx$$ I tried letting $ u = \sqrt{x} $ But I couldn't proceed. Please help.
0
votes
2answers
63 views

Can I integrate $\frac{x}{1-x}$ by substitution?

I saw a person use substitution like this: $$\int \frac{x}{1-x} dx$$ Let $u= (1-x)$, $x= 1-u, du= -1\cdot dx$ $\Rightarrow$ $-du=dx$ $$\int \frac{1-u}u (-du)$$ Can I use substitution like this? I ...