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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Integration indefinite integral of multiple functions

I need help integrating $$\frac{x}{1-\exp(-x^2/a^2)}\exp((x-u)^2/2s^2)$$ wrt $x$, where $a$ and $u$ are constants
3
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179 views

Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x $

Problem: Show that $$ \int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \,\mathrm{d}x $$ diverges. I know that there are many questions in which this problem is solved, but I want to know if my ...
3
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136 views

Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
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125 views

Integral substitution paradox

Assume $f \in L^+(\mathbb{R})$ and $x>0$. Consider the integral $$ \int_0^\infty \frac{f\left(\frac{x}{y}\right)}{y} \: dy. $$ I am trying to make the substitution $u=x/y.$ I seem to get $$ \...
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170 views

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
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115 views

Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
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41 views

Prove that $I = \int_0^{m(m+1} y_n(x)\,\mathrm{d}x$ converges and $I \in \mathbb{Q}$.

My problem is stated as follows Let $y_0(x) = x, \ \: y_1(x) = \sqrt{x}, \ \: y_{n+1}(x) = \sqrt{y_n(x) +x\,} \ $. Now define $ \displaystyle \hspace{3cm} I_n = \int_0^k y_n(x)\,\mathrm{...
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120 views

Simplifying a Vector Integral

While reading the book - Theory and Applications of Boltzmann Transport Equation by Cercignani (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \...
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88 views

Risch Algorithm for trigonometric functions

I've implemented the transcendental Risch integration algorithm for logarithmic and exponential extensions ('classical Risch'). If I want to integrate functions containing trig-functions I have to ...
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92 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = \int\limits_{0}^{\...
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141 views

Finding the integral of $\cos\theta \cdot dt$ in terms of the integral of $\sin\theta \cdot dt$

I have an integral as follows: $$\int_0^T \cos\theta\cdot dt = xT$$ where $\theta$ is a function of $t$ I also have, $$\int_0^T \sin\theta\cdot dt = y$$ I want to solve for $T$. If the ...
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116 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ \...
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76 views

Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
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90 views

Strong Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
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40 views

Evaluating a triple integral explained step by step

Evaluate: $$ \iiint_{D}\sqrt{(1-9z^2)(1-4y^2-9z^2)}\,dx\,dy\,dz$$ where $D$ is the domain: $$D: x^2 +4y^2+9z^2\le1$$ Can someone tell me if my steps are correct? $$\int_{\frac{-1}{3}}^{\frac{1}{3}} ...
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50 views

How could i find the pdf of exponential distribution from its characteristics function?

I know that the characteristics function of the exponential distribution is as following: $$ \phi_x(t) =\frac{\lambda}{(\lambda -it)}$$ Also, I know that the pdf of the exponential distribution is: $$...
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185 views

Integral vs antiderivative

I have a similar question to this one: Integrable or antiderivative. If a function has an antiderivative, does the difference of values of the antiderivative on the endpoints of an interval always ...
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415 views

Hard integration problems book, special functions

I want hard integration problems which level is college competition or harder. I want problems book about hard integration. Would you recommend some problems books? And can you recommend books ...
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228 views

Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
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44 views

If an integral over the plane vanishes, prove that it vanishes on a square.

Let $f\in L^1(\mathbb{R}^2)$ with respect to the Lebesgue measure $m\times m$ on $\mathbb{R}^2$. Prove that if $$\iint_{\mathbb{R}^2} f(x,y)dxdy=0$$ then there exists a square $S_{a,b}=\{(x,y)\,|\, ...
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36 views

How do I tackle this integral: $\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$? Is my solution correct?

I want to solve the following integral: $$\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$$ I did the following: Substitute $\gamma(k) = k-k_0 \Leftrightarrow k = \gamma + k_0;~\gamma(\pm\infty) = ...
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61 views

Integration over time by having derivation

Assume we want to find the following integration: \begin{equation}\int_{t=0}^{\infty} p(t)dt\end{equation} where $p(0)=p$ and also $$\frac{dp(t)}{dt}=-p(t)(1-p(t))\mu$$. Is there any easy way to ...
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55 views

Simplifying a Fourier integral

I have what is effectively a Fourier integral resulting from applying Abbe's theorem that I would like to simplify (ideally into a closed form solution): $$ f(\theta_0,\theta_1;\alpha) = \int_{\...
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56 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
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125 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
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359 views

Is there a generalization of integration by parts?

In the original integration by part formula there are two functions $u(x)$ and $v(x)$. What if the integral involves another function $w(x)$ as well? Second of all, I know that there is a several ...
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121 views

Prove the converse of convolution theorem

I am trying to prove the converse of convolution theorem: $$ \mathscr{F}[f(x)g(x)]=\frac{1}{\sqrt{2\pi}}\,\widetilde{f}(\omega)*\widetilde{g}(\omega)$$ I try to apply the definition of convolution ...
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119 views

An integral identity for $\frac{x^{a-1}}{x^b-1}$ via. partial fractions

Can somebody please confirm or correct the following? If $a$ and $b$ are both positive integers such that $a<b$ and $b$ is even then we can write $$\frac{x^{a-1}}{x^{b}-1}=\frac{1}{b\left(x-1\right)...
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173 views

Indefinite Integral

I tried to solve this indefinite integral $$\int\frac{1}{1+\tan^{-1}x}dx$$ I try taking the change of variable $u=\tan^{-1}x$ but I fail to reach a solution. Can anyone help me. Thanks in advance.
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74 views

Definite integral including the Chebyshev polynomial

I would like to know the proof of $$ \int_a^b \frac{T_n(x/a)T_n(x/b)\, dx}{x(b^2-x^2)^{1/2}(x^2-a^2)^{1/2}}=\frac{\pi}{2 ab}, 0<a<b, n \in \Bbb N $$ where $T_n(x)$ is the Chebyshev polynomial of ...
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Is this integral expressible in terms of generalized hypergeometric functions?

While carrying out a calculation, I encountered this integral: $$\int_0^1 d u~u^{-1-2 x} (1-u)^{-x} \Big({}_2F_1\left(1, 1, 1-x; u\right)\Big)^2\,.$$ I read in Exton's book that it is expressible in ...
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487 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ I(\epsilon)=P\int_{-1}^{\epsilon^2}\frac{\epsilon}{x\sqrt{(\epsilon^2-x)(1-x)}}...
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127 views

An interesting integral

How to integrate: $$ \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}.$$ I read about this problem on the Wikipedia Risch algorithm page, they gave an answer but I am at a loss how they got it....
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143 views

Integrate: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$

I am trying to solve the integral: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$ where $x$ is real and $a, b, n$ are positive real constants. I ...
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139 views

Attempted proof of the first part of the Fundamental Theorem of Calculus

So I was trying to prove the first part of the Fundamental Theorem of Calculus in a different way using the Riemann sum definition of the definite integral, rather than the way it was presented in my ...
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102 views

Transforming a Riemann-Stieltjes integral related to the factorial

I have been able to show that $$\log n! = \int_{1 + \epsilon}^n \log x \, d\lfloor x \rfloor$$ but I have not been successful trying to transform this Riemann-Stieltjes integral to an ordinary ...
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192 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
3
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99 views

Solving exponential integral

Any idea how to solve this integral, I tried the integration by parts, and it made the things even more difficult. The substitution didn't work either. Here is the integral: $\displaystyle \int \frac{...
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38 views

Is this a decomposition of the same function?

Let's say we have some integral, such that for a particular function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ $$\int_{\mathbb{R}^{n-m}} \int_{\mathbb{R}^m}f^+ - \int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}...
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84 views

Determine the behavior of a function defined by an integral

Suppose we have a function defined by $$\varphi(s)=\int_{-\infty}^\infty f(x,s)\,dx$$ defined for $s\in S\subseteq \mathbb{R}$. Suppose we know that it blows up at $a\in \partial S$, and we want to ...
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106 views

Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has $...
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98 views

Two properties about Bessel function

Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
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122 views

Integral $\int_{0}^\infty\frac {(1-{{e}^{-i (q-p)t}})ln(|p^2-p_0^2|)}{(q-p)({{ p}}^{2}-{{p_1}}^{2})({{p}}^{2}-{{p_2} }^{2})}dp$

I am trying to get a closed form analytic result for the integral $$\int _{0}^{\infty }\!{\frac {\left(1-{{\rm e}^{-i \left( {q}-{p} \right) t}}\right){\rm ln}(|p^2-p_0^2|)}{ ( {q}-{p} ) \left( {{ p}...
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139 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} e^{-\...
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99 views

solve nonlinear second order ODE

I obtained Nonlinear second order differential equation as $y\cdot y''+y'^2-m\cdot y^{-a}y'^2+k=0$, Where $y'= \dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$. I could not obtain the solution so please ...
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1k views

Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
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218 views

Integration of a function where the integral is equal to 0

Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function. Suppose that $\displaystyle\int_a^b x^nf(x) \, dx=0$ for all $n\in\{ 0,1,2, \ldots \}$. Prove that $f=0$. (Hint. Consider $\displaystyle\...
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72 views

infinite sum and integral representation

assuming all the numbers $ k_{n} $ are REAL is then true that $$ \sum_{n=0}^{\infty}\frac{1}{k_{n}^{s}}=\frac{1}{2 \Gamma (s) cos(\pi s)}\int_{0}^{\infty}dtt^{s-1}\sum_{n=0}^{\infty}cos(k_{n}t) $$ ...
3
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0answers
172 views

Integrating the exponential of a complex quadratic matrix

Problem statement I'm trying to do a discretized path integral/functional integral. The integral that I'm stuck with is of the form $$ \int_{-\infty}^{+\infty} \mathrm{d}^3\vec{x}_1\, \mathrm{d}^3\...
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149 views

How to integrate the following formula about normal distribution

How to compute the following formula? $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx $$ $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$ where $\Phi(x)=\int_{-\...