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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1
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0answers
46 views

what does this integral stand for?

i would really appreciate some advice concerning a paper i'm reading: http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/Leland%20port%20ins%20JF%2080.pdf on page 586, there is a problem ...
4
votes
3answers
253 views

Integration theorem: enough assumptions?

Let $f:[a,b]\to\mathbb{R}$ be a continious function. Show that if $$\int_a^b f(x)g(x)dx=0$$ for all continious functions $g:[a,b]\to\mathbb{R}$ with $g(a)=g(b)=0$, then $f(x)=0$ $\forall ...
7
votes
3answers
99 views

Is really $f(x)=\int g(x) dx$ a function?

I saw many of this kind of questions on some text/question books. Is there any other explanation of this, or is it really wrong as I thought? Here is a question of that kind: If $\displaystyle ...
1
vote
0answers
28 views

Integrating lower incomplete gamma function raised to the power $k$

I'm trying to solve the following integral: $$\int_0^\infty \gamma(t,x)^k x^t e^{-x} \mathrm{d} x$$ I'm fighting with it for quiet a while and didn't get any result. Though, I do have the ...
0
votes
2answers
22 views

Fourier function expansion for extension over a $2\pi$ period

So I am currently looking at a fourier expansion for $$f(x)=\left\{\begin{array}{ccl}\sin x &\text{ if }& x\in[0,\pi]\\0 & \text{ if } & x\in[\pi,2\pi]\end{array}\right.$$ I am ...
0
votes
0answers
25 views

prove of integral identity

I want to prove $$ \int_a^b{f(-x)dx} =\int_{-a}^{-b}{f(x)dx} $$ Here is my solution: let $u = -x; du = -dx$ $$ \int_a^b{f(-x)dx} = \int_{-a}^{-b}{f(u)(-du)} = \int_{-b}^{-a}{f(u)du} $$ And in the ...
1
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0answers
16 views

Trajectory of vector field$ v = (x^2 -1, xy)$

I have been tasked with finding the trajectory of the vector field $v = (x^2 -1, xy)$ My solution yields $x^2 - y^2 / c = 1$ However, the textbook answer gives $x^2 +- y^2 / c^2 = 1$. I ...
2
votes
0answers
29 views

How to compute the L^2-distance of a given function to the set of Gaussian functions

I am faced with the following question: given a probability density function $f$ over $\mathbb{R}$ with $\int_{\mathbb{R}}f(x)x^2dx=\sigma^2$ given, I am trying to find the "nearest" Gaussian to ...
1
vote
0answers
77 views

Relation of $\int \frac{f}{g}$ and $\frac {\int f}{\int g}$

I have two sequences of continuous functions $f_n,g_n$ on a compact space, e.g. $[0,1]^2 $. I know that $f_n,g_n>0$, that $f_n,g_n$ converge pointwise to $0$ on $[0,1]^2$, and $f_n(x,y)/g_n(x,y)$ ...
1
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0answers
14 views

compute the L^2-distance of a given function to the set of Gaussian functions

I am faced with the following question: given a probability density function $f$ over $\mathbb{R}$ with $\int_{\mathbb{R}}f(x)x^2dx=\sigma^2$ given, I am trying to find the "nearest" Gaussian to ...
4
votes
2answers
200 views

Find the smallest value of the function $F:\alpha\in\mathbb R\rightarrow \int_0^2 f(x)f(a+x)dx$

Let $f -$ fixed continuous on the whole real axis function which is periodic with period $T = 2$, and it is known that the function $f$ decreases monotonically on the segment $[0, 1]$ increases ...
1
vote
1answer
33 views

Calculate the integral $\iint_D (y^2-x^2)^{xy} (x^2+y^2)dxdy$ on a certain region

Let $D$ be the region that's bounded by $xy=a, xy=b, y^2-x^2=1, y=x$ in the first quadrant. Calculate the integral $\iint_D(y^2-x^2)^{xy}(x^2+y^2)dxdy$. Firstly, I was able to show that the boundary ...
0
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0answers
16 views

Show Fejer kernel on the real line is good, without using trignometric integrals.

This is from page 163 of Stein's Fourier Analysis. Fejer kernel on the real line is defined by $$ \mathcal{F}_R(t) = R\left(\frac{\sin(\pi t R)}{\pi t R}\right)^2$$ When $t=0$, ...
0
votes
0answers
33 views

polar coordinates : integration over negative radius

In polar coordinates , if I want to integrate the function $$\frac{2}{1+\sqrt{x^2+y^2}}$$ over the quadrant of circle that lies within $$\theta=\pi\,\,\,\, to\,\,\,\, \theta= \ 3\pi/2$$ either we do ...
-1
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1answer
17 views

Bounded convergence theorem - 2M

Can someone please help me with where the 2M is coming from?
0
votes
1answer
62 views

Riemann Sum of $f(x)=2^x$

Using Riemann Sums, how can I compute the integral $$\int_{0}^{2} 2^x dx$$ I don't know how can I take the Partition and then compute the sums , someone can help to understand this method of Riemann ...
0
votes
1answer
57 views

Can $\int_0^1 \frac{1}{x} e^{-x} dx$ be integrated?

I have an integral with a singularity at $x = 0$. $$\int_0^1 \frac{1}{x} e^{-x} dx$$ It's not a removable singularity so is it possible to perform the integration? For example could some complex ...
2
votes
2answers
69 views

Solve the differential equation:$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,dz}{ly-mx}$

QUESTION: Solve the differential equation: $$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,dz}{ly-mx}$$ MY ATTEMPT: I tried out to proceed by using ...
0
votes
2answers
30 views

How can we find the bounds of the following integrals?

How can we find the bounds of the following integrals? $$\int_{0}^{+\infty}\int_{0}^{+\infty}\phi(x,y)dxdy$$ where $\phi(x,y)=$ $\begin{cases} 1 \quad if \quad x=y>0 \\ 0 \quad otherwise ...
0
votes
2answers
70 views

Integrate $ \int \frac{4x^2+2x}{(3x-1)(x^2+1)}dx$.

$$ \int \frac{4x^2+2x}{(3x-1)(x^2+1)}dx$$ I am having difficulty determining which integration technique to use for the above question. I have tried partial fraction decomposition with two ...
1
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2answers
57 views

Determine the value of c that makes the blue area above y = c equal to the blue area below y = c.

Determine the value of c that makes the blue area above y = c equal to the blue area below y = c. edit: I'm kind of stuck on this problem, not sure what steps to do so that I can find the equal ...
2
votes
1answer
55 views

Evaluating $\int^1_0 x^n \sqrt{1+x} dx$ for $n \in \mathbb{N}$

As in the title, my question is very simple: how can I evaluate: $$ I_n = \int_0^1 x^n \sqrt{1+x} \;dx $$ where $n \in \mathbb{N}$. Seeing as n is a natural, I feel like one way I could approach ...
2
votes
2answers
99 views

Integral of $e^{\cos t}$

I’d like help with computing the following integral: $$\int_0^\pi e^{\cos t}\,dt.$$ (This is a problem in complex analysis [supposedly].)
2
votes
1answer
36 views

Problem solving $\int_{1}^{2}\int_{x}^{2x}\int_{\sqrt{1-x^{2}-y^{2}}}^{\sqrt{2xy}}\frac{zdzdydx}{x^{2}+y^{2}+z^{2}} $

Good night, i have a problem solving this integral: $$\int_{1}^{2}\int_{x}^{2x}\int_{\sqrt{1-x^{2}-y^{2}}}^{\sqrt{2xy}}\frac{zdzdydx}{x^{2}+y^{2}+z^{2}}$$ I think make a change to spherical ...
0
votes
1answer
27 views

Integrals with $e^z$ [on hold]

I need help starting this question. Calculate the integral of $$f(x,y,z)=e^z$$ over the portion of the plane $$x+2y+z=4$$ where $x,y,z$ is greater than or equal to 0.
0
votes
1answer
37 views

Problem with a limit with a integral in it

Suppose that the temperature in a long thin rod placed along the $x$-axis is initially $\frac{C}{2a}$ if $|x| \leq a$ and $0$ if $|x| > a$. It can be shown that if the heat diffusivity of the rod ...
0
votes
1answer
13 views

Triple Integral of 6xy dV, why is lower bound of z 0 instead of 1?

In the Stewart Calculus Textbook 7th Edition, problem 13 in chapter 15.7 states: "Evaluate the triple integral: Triple Integral within E of 6xy dV, where E lies under the plane z=1+x+y and above the ...
1
vote
1answer
45 views

How can I calculate $\int_1^2 \int_{\sqrt{x}}^x \sin\left(\frac{\pi x}{2y}\right) \,dy \, dx$?

Good night i have problem solving this integral. $$\int_1^2 \int_{\sqrt{x}}^x \sin\left(\frac{\pi x}{2y}\right) \,dy \, dx$$ I make the area of integration, but i cannot solve the integrat, i don't ...
0
votes
0answers
22 views

Name or reference about a inequality with integrals?

I have wrote down some class notes and I think I copied something wrong. It is an integral inequality; $$\iiint_{B^n}|\nabla\psi|^2\frac{1}{|x|^{n-2}}dV\leq C\iint_{\partial B^n}|\psi|^2dA$$ where ...
1
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0answers
36 views

the sum-int exchange

Why is the sum-int exchange allowed in the following equality ($f$ is a $C^1$ function): $$ \sum_{m\geq 1} \int_0^{\pi}f'(t) \frac{\sin((2m+1)t)}{2m+1} dt = \int_0^{\pi}f'(t) \sum_{m\geq ...
0
votes
1answer
32 views

The reason of $\int_{-\infty}^{\infty}\mu_k^2(x)dx=1$

Is there anyone could tell me why if $$\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}}=\int_{-\infty}^{\infty} (\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}} \mu_k^2(x))dx= \sum_{k \geq 0} e^{it ...
0
votes
2answers
36 views

Relation between $\int_{a}^{b} f(x) dx$ and $\int_{a}^{b} (1-f(x)) dx$

Say you're expected to work out $\int_{0}^{\pi/3} \sin^2(x) dx$ solely from the result $\int_{0}^{\pi/3} \cos^2(x) dx$. It can be transformed into $\int_{0}^{\pi/3} (1-\cos^2(x)) dx$, but then what?
14
votes
1answer
360 views

Why does the hard-looking integral $\int_{0}^{\infty}\frac{x\sin^2(x)}{\cosh(x)+\cos(x)}dx=1$?

I have to ask this question; most looking complicated definite integral yield not so nice closed form or irrational numbers or mixed of what ever ect. Why is this particular hard looking integral ...
5
votes
3answers
102 views

Solution of integral $\int \frac{\sin (x)}{\sin (5x) \sin (3x)} dx$

Find the following integral $$\int \frac{\sin (x)}{\sin (5x) \sin (3x)} dx$$ I don't know how to deal with the $\sin (x)$ in the numerator. If it had been $\sin (2x)$ then we could have used $\sin ...
0
votes
2answers
27 views

Generating definite integral for unit circle without consulting tables

I realize that I can consult integration tables for integrals like this, but I wanted to test my knowledge of integration techniques to solve this integral for the area of the unit circle. ...
1
vote
2answers
23 views

expressing contour integral in different form

Hi I have a short question regarding contour integration: Given that $f(z)$ is a continuous function over a rectifiable contour $z = x + iy$. If $f(z) = u(x,y) + iv(x,y)$, why does it follow that the ...
0
votes
0answers
21 views

Integral of convex function applied on a function

Let $f$ be an integrable function of $\mathcal{L}(\mathbb{C},\mathbb{R})$, measure Lebesgue. I want to prove that there exists an increasing convex function $H:\mathbb{R}^+\rightarrow\mathbb{R}^+$ ...
0
votes
1answer
30 views

Definite Gaussian/exponential integral

I've been revising some quantum mechanics, and I was wondering how I would calculate a specific standard deviation. The wave function I am working with is ...
-2
votes
0answers
24 views

integration of a non-negative continous functions

Let $f:[0,1] -> \mathbb R$ be continuous such that $f(t) \geq 0$ for all $t$ in $[0,1]$. Define $g(x) = \int_{0}^{x}f(t)dt$ then a) $g$ is monotone and bounded. b) $g$ is monotone, but not ...
0
votes
0answers
22 views

Integral computation with Mathematica and Sympy differ

To compute the integral: $I = \int_{0}^{+oo} ue^{Au^{2}+Bu}du$ where $A<0$ and $B>0$ I have tried both Mathematica and Sympy but they yield different results: Mathematica yields: $ I = ...
0
votes
0answers
80 views

If $ \int_{0}^{2}\frac{ax+b}{(x^2+5x+6)^2}dx = \frac{7}{30}\;,$ Then value of $a^2+b^2$

If the value of Definite Integral $\displaystyle \int_{0}^{2}\frac{ax+b}{(x^2+5x+6)^2}dx = \frac{7}{30}\;,$ Then value of $a^2+b^2$ $\bf{My\; Try::}$ We can write it as ...
1
vote
2answers
30 views

Simplification idea for finding antiderivative

Is there a simple way of finding the anti-derivative $F$ (i.e. $F(x)=\int f(x)dx$) of $$f(x)=\frac{1}{(\sqrt{x}-1)\sqrt{x}}$$ I've managed to do it by 2 by parts integrations in row, but that took ...
3
votes
2answers
63 views

If $f(x) = ax^2+bx+c$ and $f(0) = 0$ and $f(2) = 2\;,$ Then minimum value of $\int_{0}^{2}|f'(x)|dx$

Consider the polynomial $f(x) = ax^2+bx+c$ and $f(0) = 0$ and $f(2) = 2\;,$ Then Minimum value of $\displaystyle \int_{0}^{2}|f'(x)|dx$ $\bf{My\; Try::}$ From $f(0) =0\;,$ We get $c=0$ and ...
2
votes
1answer
47 views

Where is my mistake calculating $\int_{-\infty}^{\infty}\frac{x\sin(x)}{x^2+4}~dx$?

Where is my mistake calculating $$\int_{-\infty}^{\infty}\frac{x\sin(x)}{x^2+4}~\text{d}x$$ Let $$f(z)=\frac{z\sin(z)}{z^2+4}$$ it has simple poles at $\pm 2i$. We take the standard half circle path ...
2
votes
3answers
99 views

Real Analysis question on FTC, Integral

Let $g:[0,1] \rightarrow \mathbb R$ be a continuous function and assume that $$ \int_{0}^{1} g(x) \phi'(x) dx = 0 $$ for all continuously differentiable functions $\phi: [0,1] \rightarrow ...
1
vote
0answers
24 views

Use double integrals to find the volume of the solid obtained by the rotation of the region

I am unsure how to set up the integrals. The integration should be done in polar coordinates. Here is what is give: $D = \{ (x,y,z) | x^2 \le z \le 6 - x, 0\le x \le 2, y = 0\}$ in the xz-plane ...
0
votes
2answers
75 views

Is $\frac{1}{x^2}$ Lebesgue integrable while $\frac{1}{x}$ is not?

My textbook defined integrability as $f$ is said to be Lebesgue integrable if $\int{}f$ is finite. I heard that $\frac1x$ is not Lebesgue integrable, but $\frac{1}{x^2}$ is Lebesgue integrable. I do ...
4
votes
1answer
58 views

How do you show that $\int_{0}^1\frac{dx}{x^x}=\sum_{k=1}^\infty\frac{1}{k^k}$?

My task is this: i) Find the sum to$$1-x\ln x +\frac{1}{2}(x\ln x)^2-\ldots+\frac{(-1)^k}{k!}(x\ln x)^k+\ldots$$ (ii) The great norwegian mathematician Atle Selberg showed that ...
1
vote
1answer
46 views

Limit of a Riemann sum: $\lim_{n\to\infty} {n^5 \sum^n_{r=0}\frac1{(n^2+r^2)^3}} $

Required to find $\lim_{n\to{\infty}} {n^5 \sum^n_{r=0}\frac{1}{(n^2+r^2)^3}} $ $\lim_{n\to{\infty}} \frac{1}{n} \sum^n_{r=0}(\frac{n^2}{n^2+r^2})^3$ $\lim_{n\to{\infty}} \frac{1}{n} ...
0
votes
0answers
35 views

How to integrate $\cos2\pi\left(x+\frac{n}{x}\right)$

This is a follow up question of Integrate $\cos^2(\pi x)\cos^2(\frac{n\pi}{x})$. By using product to sum formula, this could be converted to question to integrate ...