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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17 views

Finding a function $f: \mathbb{R} \rightarrow \mathbb{R}$ with this property?

Find a function $f: \mathbb{R} \rightarrow \mathbb{R}$ that is Riemann integrable on every bounded and closed interval, such that the function $$ g: \mathbb{R} \rightarrow \mathbb{R}: x \mapsto \...
1
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0answers
33 views

A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...
-3
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0answers
18 views

can any one help me with this integral please

$$\int_0^\infty y \exp \left[ -\frac{y}{c} - Ly^{\frac{m}{2} -1}\right]dy$$ where: $c$, $m$ and $L$ are constants.
3
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1answer
31 views

Integration of Exponential of Gaussian [duplicate]

I am interested in the following integral $$\int_{-\infty }^{\infty } \left[1-\exp\left(-\frac{e^{-\frac{x^2}{2\sigma^2}}}{\sqrt{2 \pi\sigma^2 }}\right)\right] \, dx$$ Does any one know if an ...
0
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1answer
25 views

Evaluation of definite Integral Containing Rational functions..

Evaluation of $$\int_{-5}^{-2}\left(\frac{x^2-x}{x^3-3x+1}\right)^2dx+\int_{\frac{1}{6}}^{\frac{1}{3}}\left(\frac{x^2-x}{x^3-3x+1}\right)^2dx+\int_{\frac{6}{5}}^{\frac{3}{2}}\left(\frac{x^2-x}{x^3-3x+...
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3answers
55 views

Final step in integration

I was solving a integration problem in which we have to integrate the following $$\int\frac{dx}{x\sqrt{x^2+1}}$$ I tried it lot , i think almost got the last step but the answer did not match . My ...
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1answer
58 views

How to compute integral $\int_0^1\sqrt{1-x^2}dx$ without using trigonometric functions and also answer should not contain trigonometric functions

I am working on definite integrals, and not getting solution for above question, anyone can help me in this?
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0answers
23 views

How do I justify this differentiating under the integral? (Complex “Wirtinger derivative”)

I would like to differentiate under the integral in this situation: $$\frac{d}{d\bar{z}}\int_0^\infty h(t)e^{tz}\,dt=\int_0^\infty \frac{d}{d\bar{z}}h(t)e^{tz}\,dt$$ where $\frac{d}{d\bar{z}}$ is ...
1
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2answers
80 views

Does $\int^{\infty}_0 \frac{\cos x}{1+x}\,\mathrm dx$ converge?

The integral $$\int^{\infty}_0 \frac{\cos x}{1+x}\,\mathrm dx$$ is shown to be equal to $$\int^{\infty}_0 \frac{\sin x}{{(1+x)}^2}\,\mathrm dx$$ through integration by parts. The latter one converges ...
1
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1answer
43 views

Integral substitution?

In a recent economics paper (Rhodes and Wilson 2016), expected buyer surplus is computed as follows (page 8, (4)): $$ v^*(q) = \int_{p^*(q)}^{b+q}[1-G(z-q)] dz$$ where $G(\cdot)$ is a distribution ...
2
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0answers
30 views

Finding the boundaries on a triple integral

Solve: $$\iiint yz \,dV$$ Over the tetrahedron with vertices on the points $$A(0,0,0), B(1,1,0), C(1,0,0), D(1,0,1)$$ Well, I proceeded to find a the equation of a plane which contained B, C and D. ...
0
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1answer
42 views

F(x) = 0 for all points except c. Show F is integrable.

Suppose $c$ is a point in the closed $[a,b]$ and that $F(x) = 0$ for all $x$ in $[a,b]$ except for $c$ and that $F(c) = 1$. Show that $F$ is integrable on $[a,b]$ and that $\int_a^bF(x)dx = 0$. By ...
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0answers
19 views

I have a problem I would like some advice with. I have to discretize bellow integration over volume in 2D.

I know what is discrete form of $$\int_\Omega\nabla\phi\,dV$$ in which $\phi$ is vector in 2D and $\Omega$ is volume of cell in CFD field. The result is: $$\frac{1}{\text{Volume}}\sum_{\text{face}} n\...
10
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4answers
751 views

Calculate the following integral

$$\int_{[0,1]^n} \max(x_1,\ldots,x_n) \, dx_1\cdots dx_n$$ My work: I know that because all $x_k$ are symmetrical I can assume that $1\geq x_1 \geq \cdots \geq x_n\geq 0$ and multiply the answer by $...
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1answer
24 views

Let $f \in L^1(\mathbb{R}), F(x)=\int_{-\infty}^x f(y) \text{d}y, F \in L^1(\mathbb{R})$ then $F$ vanishes at infinity

Good day, Currently I am working with the book "Introduction to Fourier Series" by R. Lasser. There we have the following theorem (on page 227): Let $f \in L^1(\mathbb{R})$. Set $$F(x):=\int_{-\...
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0answers
13 views

Elaboration on Lagrange's algebraic approach to calculus?

In lectures 4 and 8 of NJ Wildberger's differential geometry series, he presents an algebraic method of taking the derivative of a polynomial that was apparently devised by Lagrange. He briefly ...
1
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0answers
22 views

Conditional mutual information for continuous random variables

Cover and Thomas provides definition of Conditional Mutual Information (CMI) for discrete random variables but doesn't say anything about continuous variables. Wikipedia has a section about a "more ...
6
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3answers
85 views

Evaluation of $\int^{\pi/2}_{0} \frac{x \tan(x)}{\sec(x)+\tan(x)}dx$

Evaluate the given integral: $$\int^{\pi/2}_0 \frac{x \tan(x)}{\sec(x)+\tan(x)}dx$$ I multiplied and divided by $\sec(x)+\tan(x)$ to get denominator as $1$ but In calculation of integral, $x$ is ...
2
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5answers
176 views

Evaluating $\int \sqrt{x^2-3}\:dx$

I need to solve: $$\int \sqrt{x^2-3} \, dx.$$ So I use the substitution: $$x=\frac{\sqrt 3}{\cos(t)}$$ $$dx= \frac{\sqrt 3 \sin(t) \, dt}{\cos^2(t)} $$ and I get $$3\int \frac{\sqrt{\frac1 {\...
1
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1answer
27 views

Fourier coefficient and computing an improper integral

I am having difficulties with this problem. I don't really know where to start, I suspect there is something I am supposed to know or "see" that I am missing. $u$ is a $2\pi$-periodic function ...
2
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1answer
44 views

Basic question about improper integral

If I have the evaluate $\int_{-1}^{1} \frac{1}{x^3}dx$ can I solve it by evaluating $$\lim\limits_{a \to0^{-}}\int_{-1}^{a}\frac{1}{x^3}dx+\lim\limits_{b\to 0^{+}}\int_{b}^{1}\frac{1}{x^3}dx~\qquad ?$$...
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1answer
49 views

Is there any other possibility

$$\int f(x) dx =f(x)$$ Then $$\int \left( f(x)\right)^2 dx$$ is equal to I know that $e^x$ will satisfy this . Is there any other function that will satisfy this or $e^x$ is the only one which ...
0
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1answer
22 views

Surface element for a cylinder, how?

I having problem how to find the differential surface element for a cylinder $x^2+y^2=r^2$ with height $l$. The surface have three parts; top, cylinder and bottom. I know how to parametrize and ...
8
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4answers
648 views

Are there any conditions of integration?

When we differentiate a function $f(x)$, there are conditions under which the derivative would not exist and cannot become differentiable. However, I have tried looking online for any conditions for ...
6
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6answers
166 views

Integrating $\int \dfrac{1+x^2}{1+x^4}dx$

I am trying to integrate this function, which I got while solving $\int\frac{1}{\sin^4( x) + \cos^4 (x)}$: $$\int \dfrac{1+x^2}{1+x^4}dx$$ I think to factorise the denominator, and use partial ...
0
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0answers
18 views

How many integrals to add to find integral of circle with another circle touching internally?

Here Would it be three or more integrals? I think it is three, but I am not sure because then the upper arc of the smaller circle is not included.
2
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2answers
69 views

Integrating square root with condition

I am an engineer working on a problem that requires the use of integration to calculate compression force within a segment. I have worked out the formula, I just need help with the integration as I ...
1
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2answers
58 views

integral solution needed for general powers of $x$

I want to find the solution to the following integral $$\int_0^{\infty} \frac{dx}{x^{\frac{\alpha}{2}}+1}$$ where $\alpha$ can be any value greater than 2 such that $\alpha /2 >1$ but can be any ...
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3answers
88 views

Is it possible to $\int \sqrt{\cot x}$ by hand

$$\int \sqrt{\cot x}{dx}$$ $$\int \sqrt{\frac{\cos x}{\sin x}}{dx} $$ Using half angle formula $$\int \sqrt{\frac{1-\tan^2 \frac{x}{2}}{2\tan \frac{x}{2}}}{dx}$$ But I am not getting any lead from ...
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0answers
25 views

When an inner product of two continuous functions and one of them are given, how can I find the other one?

$f(p)$ is the inner product of $h(t,p)$ and $x(t)$. That is, $\int_{T} h(t,p)x(t)dt = f(p)$ When $h(t,p)$ and $f(p)$ are given, how can I find $x(t)$? $0 <= t < T$ and $0<= p < \inf$
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3answers
75 views

How to find a derivative of $f(x)=\int_0^{x^2}e^{xt^{-2}}dt$

Let $$f(x)=\int_0^{x^2}e^{xt^{-2}}dt$$ I want to find $$f'(x)$$ I tried to use taylor expansion: $$e^{xt^{-2}}=\sum_{n=0}^\infty \frac {x^nt^{-2n}} {n!}$$ Indefinite integral gives, $$\int e^{xt^{-2}}...
1
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1answer
40 views

Help needed in solving integration

I want to solve the following integration $$\int_0^{\infty}[1-(\frac{1}{1+x})^M]x^{-\frac{2}{\alpha}-1}dx$$ where $M$ is a positive integer and $\alpha \geq 2$ My attempt: In my attempt I use the ...
3
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2answers
53 views

What is the geometrical meaning of the integral of a vector valued function?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is an integrable function. then $\int_a^b f(x)dx$ can be considered as the area between the graph and the x-axis. But what if $f:\mathbb{R}^n\rightarrow \...
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0answers
24 views

If $f(x)\geq 0$ for all $x \in [a,b]$ and $\alpha \in BV([a,b])$ is increasing , then $\int_a^bf d\alpha \geq 0.$

This is a proof verification question. Here, $\, f$ is continuous and $\alpha$ is of bounded variation. My only tools are the sums, for a given partition $P = \{a=x_0 < \ldots < x_n = b \}$ of $...
3
votes
3answers
130 views

$\int f(x)\,dx - \int f(x)\,dx$

which is true $$\int f(x)\,dx - \int f(x)\,dx = 0$$ or $$\int f(x)\,dx - \int f(x)\,dx=c\text{ ?}$$ with $c$ some arbitary constant. My intuition says that 'something' subtracted by itself is ...
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1answer
51 views

Prove the following integral identity

Does any one have any idea on how to prove the following: $$\int_0^Sf(z)\,\mathrm{d}z+a\int_0^Sf(z)\left(\int_0^zf(z_1)\,\mathrm{d}z_1\right)\,\mathrm{d}z+a^2\int_0^Sf(z)\left[\int_0^zf(z_1)\left(\...
0
votes
1answer
49 views

Solution to the convoluted integral equation

A have the following equation: $$f(a)=\int_0^ag(x)f(x)\,dx,$$ where $g(x)$ is a known function. Is there any solution to $f(a)$ just in terms of $g$?
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1answer
41 views

How to solve the following result in integration?

I want to prove the following inequality: $$\int_{a}^{b}{|\cosh(\sqrt{iw})|^2dw} \geq \int_{a}^{b} \sinh^2\big(\sqrt{\frac{w}{2}}\big) dw$$ How to solve above integral? I am unable to keep $\iota (...
2
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2answers
90 views

How to evaluate $\int\frac{dx}{(2\sin x+\sec x)^4}$?

I tried a lot but I am not able to get a start. Can anyone give me the start of this question $$ \int\frac{dx}{(2\sin x+\sec x)^4} \ ? $$
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2answers
26 views

If an integrable function is orthogonal to all derivatives, then is f a constant?

Suppose that I have a function in $f \in L^1(\mathbb{R})$ such that $$\int_{\mathbb{R}}f(x)v'(x)\,dx = 0$$ for all test functions $v$ which are smooth with compact support. Can I show that $f(x)$ is ...
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0answers
34 views

How to calculate limits in triple integral?

I have the next excersise of triple integrals: $f(x,y,z) = \cos(x + y + z)$, Limited by planes $x=\pi, y=\pi, z=\pi $ I have to find/determine the value for this. Specific Questions: How can I ...
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0answers
26 views

Is it possible construct a well-defined series through integration by parts?

I have a function $$G=\int_0^S\mathrm{d}x\,f(x)g(x)\mathrm{e}^{-\int_0^x\mathrm{d}z\,g(z)}~.$$ If $f(x)$ was unity, the above integral could have been easily written as $$G=1-\mathrm{e}^{-\int_0^S\...
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0answers
23 views

Obatin $\int_{\gamma_1}F\cdot dl =\int_{\gamma_2}F\cdot dl$

Let $F = (F_1,F_2)$ be a $C^1$ vector field such that all its components are continuously differentiable in $\Omega$. Assume that $\frac{\partial F_1}{\partial y}=\frac{\partial F_2}{\partial x}$ Let ...
2
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1answer
44 views

Hadamard-like complex variable substitution

\begin{align} \frac\pi a &= \int_{-\infty}^\infty dxdye^{-a(x^2+y^2)}\\ \tag{1}&= \int_{-\infty}^\infty dxdye^{-a(x+iy)(x-iy)} \end{align} So far so good. Now introduce a complex variable $z$ ...
4
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1answer
44 views

Volume in zero dimensional space

Suppose $A\subset \mathbb{R}^n$ is a compact, convex and centrally symmetric set such that $(x_1,\ldots,x_n)\in A$ if $$ |x_1|+\ldots+|x_r|+2\left(\sqrt{x_{r+1}^2 + x_{r+2}^2} + \ldots + \sqrt{x_{n-1}^...
3
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1answer
31 views

Convergence of a integral for every curve in the sphere

Let $S$ be the unit open sphere in $\mathbb{R}^3$: $x^2+y^2+z^2< 1$ and $\partial S$ its border $x^2+y^2+z^2= 1$. Let $f:S\cup \partial S\rightarrow \mathbb{R}$ be a continuous function which is ...
0
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2answers
49 views

Evaluate $\int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }}$

How do you solve this integral $$ \phi = \int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }} $$ ? Note: It appears in the Kepler problem and it should ...
0
votes
1answer
23 views

Choice of the limits for multivariable integral

Let $A \subseteq \mathbb{R}^2$ a limited set bordered through $x=0, x=1, y=-1+x, y=1-x^2$. Rotate A around the y-axis and define this set with $B$. Calculate the integral $$\int_B y\,\mathrm{d}x\...
1
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1answer
34 views

Why do I get two answer when calculating this integral from two ways?

Assuming $a(t)=a_0\sin(\omega t)$, $v(0)=0$ and $x(0)=0$. I hope you know about basic relation between position, velocity and acceleration. They are derivatives of the proceeding one. I went on ...
0
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2answers
37 views

Power Rule for Indefinite Integrals

To prove $\int x^p \, dx = \frac{x^{p+1}}{p+1} + C$, my calculus textbook writes: $$F '(x) = \frac{d}{dx} \left(\frac{x^{p+1}}{p+1} +C\right) = \frac{d}{dx} \left(\frac{x^{p+1}}{p+1}\right)+\frac{d}{...