Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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0
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1answer
30 views

Evaluate the integral (via infinite sum)

The following was a problem I ran into while preparing for an upcoming exam: Evaluate $\lim_{n\to \infty}\int_0^{n^{1/3}}(1-\frac{x^2}{n})^n dx$. Write in terms of a definite integral then evaluate ...
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3answers
47 views

Cooling tea: average temperature from derivative

I couldn't get the correct answer for this question: A science geek brews tea at $195 \:\rm °F$ and observes that the temperature $T(t) \:\rm °F$ of the tea after $t$ minutes is changing at the ...
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1answer
68 views

Integrate the Circumference of an Ellipse to Find the Area

For a circle of radius R, one can find the area by integrating the circumference equation in the interval $(0, R)$, $$\text{Area} = \int^R_0 2\pi r\ dr = \pi R^2$$ My intuition for this is that we'...
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1answer
56 views

Are there multiple ways to solve the integral $\int \cos^2(x)dx$

Both with the internet, places like wolfram and symbolab and in a previous question: How do you find singular solutions to first order differential equations? I get different answers for the integral ...
2
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1answer
58 views

Prove that $12(a\sin a+\cos a-1)^2\le 2a^4+a^3\sin(2a)$,$\forall a\in (0,\infty)$

Prove that $$12(a\sin a+\cos a-1)^2\le 2a^4+a^3\sin2a,\forall a\in (0,\infty).$$ The solution in the book where I found this goes like this : from CS for integrals we have that $$\left(\int_0^a x\cos ...
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0answers
19 views

Laplace transform to solve fredholm eqn

$$u(x)-\lambda\int_0^{a}tu(t)dt=f(x)$$ I want to solve generally for u(x) where $\lambda$ and $a$ are parameters and $f(x)$ is given. I have used Laplace transform to get the equation down to: $$U(s)-...
4
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1answer
116 views

Improper Fourier transform

The most common way to verify if the Fourier transform of a function $f$ is integrable $(\hat f\in L^1(\mathbb{R}))$ is by proving that the function $f$ is integrable and $f'$, $f''$ are also ...
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1answer
68 views

Closed form of $\int_{0}^1 \operatorname{Li}^2_n(x) dx$ for positive integer $n$

As stated in the title, I want to find $$I(n)=\int_{0}^1 \operatorname{Li}_n^2(x) dx$$ where $n$ is a positive integer. Using integration by parts, we have that $$I(n) = \operatorname{Li}_n^2(1)-2\...
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0answers
14 views

Multivariable function Integrable for what values?

For what values of $\alpha,\beta \in \mathbb{R}$ will the function $$f:(0,1]\times(0,1] \to \mathbb{R}: (x,y) \mapsto x^\alpha y^\alpha (x+y)^\beta$$ be integrable? Normally, I don't have problems ...
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0answers
57 views

Implication of finiteness of integral

Consider the measure space $(A,\mathcal{F},\mu)$. We say that a real measurable function $f$ on $A$ is integrable if $\int_A \mid f\mid d\mu < \infty$. Furthermore, an integral of a real ...
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2answers
330 views
+100

Challenging integral: Evaluate $\int_0^1\frac{\ln^3(1-x)\operatorname{Li}_3(x)}{x}\ dx$

How to evaluate $$I=\int_0^1\frac{\ln^3(1-x)\operatorname{Li}_3(x)}{x}\ dx\ ?$$ I came across this integral $I$ while I was trying to compute two advanced sums of weight 7. The problem with my ...
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0answers
31 views

Does $\int_V f(s)ds=\int_U f(\varphi (t))|\det \varphi '(t))|dt$ hold if $f$ is measurable?

Let $\varphi :U\to V$ a diffeomorphism where $U,V$ are open. If $f:\mathbb R\to \mathbb R$ is a Borel function integrable function, we have that $$\int_V f(s)ds=\int_U f(\varphi (t))|\det \varphi '(t)|...
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2answers
30 views

How to evaluate $\int_{1-\sqrt{1-t^2}}^{1+\sqrt{1-t^2}}(y^2-2y+t^2)dy$, where t is constant

I need to evaluate the following one. Can't understand the method in my textbook. $$\int_{1-\sqrt{1-t^2}}^{1+\sqrt{1-t^2}}(y^2-2y+t^2)dy$$ My textbook is to let $\alpha=1-\sqrt{1-t^2}$, $\beta=1+\...
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0answers
5 views

Derivation of Search Gradients for Natural Evolution Strategies

My question refers to chapter 2 of the Natural Evolution Strategies paper. I want to understand the derivation of the search gradients. Given z, a solution vector sampled from a probability ...
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1answer
43 views

Why is my derivation of the variance of a normal distribution incorrect?

I have seen a proof of the variance of a standard normal distribution, but am getting an incorrect answer am and not sure why. Here is my reasoning. $$\mathbb{V}[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^...
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0answers
18 views

Saddle point method for a stationary subspace of dimension $> 0$

I recently asked a question on Physics SE regarding the validity of using the saddle point technique when the saddle point is not only degenerate, but forms a continuous subspace of the parameter ...
4
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0answers
279 views

How can one show the multiple integral $\int_{0}^{a} \int_{0}^{a} \frac{dx ~ dy}{(a^2+x^2+y^2)^{3/2}}= \frac{\pi}{6a}$ by hand? [duplicate]

Mathematica gives the value of the multiple integral $$ \int_{0}^{a} \int_{0}^{a} \frac{dx ~ dy}{(a^2+x^2+y^2)^{3/2}}=\frac{\pi}{6a}. $$ See also this result from WolframAlpha: The question is ...
13
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3answers
1k views

Methods to evaluate $ \int _{a }^{b }\!{\frac {\ln \left( tx + u \right) }{m{x}^{2}+nx +p}}{dx} $

Today I saw a question with an answer that made me rethink of the following question, since it's not the first time I try to find an answer to it. If you look at the answer of Mhenni Benghorbal here ...
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2answers
37 views

Why are some improper integrals convergent and others divergent?

The integral of the function $f(x)=1/x^2$ is convergent and it equals 1 when the limits of the integral is $\int_1^\infty$ but it's divergent and equals $\infty$ when the limits are $\int_0^1$. I know ...
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0answers
13 views

Fubini's theorem for Borel functions with respect to Lebesgue measure

Is the following version of Fubini's theorem true? All the following integrals are with respect to Lebesgue measure. Suppose $f(x,y):\mathbf{R}^n \times \mathbf{R}^m=\mathbf{R}^{n+m}\to \mathbf{R}\...
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1answer
41 views

Calculate $\int\sqrt{(x^2+1)}dx$

Calculate $I$ =$\int\sqrt{(x^2+1)}dx$ I have tried calculating it using integration by parts: $$f'(x) = 1, f(x) = x$$ $$g(x) = \sqrt{x^2+1}, g'(x) = \frac{x}{\sqrt{x^2+1}}$$ $$\int\sqrt{x^2+1}dx = x\...
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0answers
47 views
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1answer
62 views

Evaluate $\int_0^{\infty } \frac{\sinh (a x) \sinh (b x)}{(\cosh (a x)+\cos (t))^2} \, dx$

From Gradshteyn&Ryzhik 3.514.4 we know that $$\int_0^{\infty } \frac{\sinh (a x) \sinh (b x)}{(\cosh (a x)+\cos (t))^2} \, dx=\frac{\pi b \csc (t) \csc \left(\frac{\pi b}{a}\right) \sin \left(\...
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1answer
45 views

How to integrate $\exp(-|xy| - \phi(x^2+y^2))$?

I am trying to integrate this function: $$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp\{ -\lvert xy \rvert - \phi(x^2 + y^2) \} dx dy $$ A little bit of background: the negative logarithm of ...
1
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1answer
53 views

How should I setup the integral to calculate surface area

Find the surface area of the part of the surface $y^2 +z^2 =2z$ cutoff by the cone $x^2 = y^2+z^2$ So the given surface is actually a circle with radius $1$ and centre $(0,1)$ in y-z plane,but I am ...
0
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1answer
24 views

Determine limit using DCT

I want to determine the following limit (for $\alpha>0$) $$ \lim_{\alpha \to +\infty} \int_0^{+\infty} \frac{1}{\sqrt{x}(1+x^\alpha)}dx $$ (call this integrand the fuction $f_\alpha$). Now, this ...
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0answers
28 views

Laplace Transform of e^{-at}

I have some trouble understanding the Laplace Transform of $x(t) = e^{-at}u(t)$, where $u(t)$ is the Heaviside step function. When calculating the integral, we get to a point where $$ X(s) = \int_{0}^...
13
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4answers
320 views

Integration with $\ln(x)$ in the denominator

Find$$\displaystyle\int_1^\infty\frac{(x^2-1)(x^4-1)(x^6-1)}{\ln(x)(x^{14}-1)} dx$$ I tried simplifying the terms without logarithm $x^2-1=(x-1)(x+1)\\x^{14}-1=(x^7-1)(x^7+1)$ to see if any ...
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0answers
19 views

Controllability Gramian via integration by parts

I'm trying to program something like this article. On page 3, there's the following equation (eq. 6) that should be expressible in closed-form: $$\int_0^te^{(A(t-t'))} M e^{(A^T(t-t'))}dt'$$ where $...
3
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2answers
52 views

Stopping Car Question

Suppose that as a yellow car brakes, its velocity is described by $$v(t)=3.3e^{1-t}-0.6$$ If the brakes are applied at time t=0 seconds, what is the distance it takes for the car to come to a ...
11
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3answers
408 views

Proving $\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx=\frac{3 \pi ^2}{160}$

How to show that the integral $$\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx$$ equals to $\frac{3 \pi ^2}{160}$? I've already verified this numerically ...
24
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5answers
2k views

What's the “limit” in the definition of Riemann integrals?

Consider one of the standard methods used for defining the Riemann integrals: Suppose $\sigma$ denotes any subdivision $a=x_0<x_1<x_2\cdots<x_{n-1}<x_n=b$, and let $x_{i-1}\leq \xi_i\...
3
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0answers
75 views

$\int \log(x+e^x) \mathbb{d}x$, or When every CAS fails

No computer algebra system -- at least to my knowledge -- managed to either compute the integral $\int \log(x+e^x)\space \mathbb{d}x$, in terms of any known functions, or even just prove that it is ...
8
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3answers
391 views

Seemingly impossible double integral reduction

Show that $\int_{0}^{1}\int_{0}^{1}(xy)^{xy}dxdy = \int_{0}^{1}y^{y}dy$ I have tried the method used in the Gaussian integral, polar coordinates, exp(.) and ln(.) of both sides of the integrand... ...
8
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3answers
236 views

Is this closed-form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?

According to Freitas' paper at page $11$. $$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)-\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$ I evaluated the LHS and it is $0....
4
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1answer
70 views

Proving the existence of a continuous function with same integral as another function

Let f be a Riemann-integrable function in $[a,b]$. I have to prove that $\forall \epsilon >0 , \exists g$ continuous , such that $ g \leq f$ and $\int_a^b f - \int_a^b g < \epsilon $. I thought ...
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2answers
37 views

Orthogonal complement of closed subset in Hilbert space

Consider a subset $K\subset L^2 (\mathbb{R})$ defined by $$ K=\{f\in L^2(\mathbb{R})\mid \forall n\in\mathbb{Z}: \int_{n}^{n+1}f(x)dx=0\} $$ I want to determine the orthogonal complement $K^\bot$ in ...
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0answers
17 views

How do we calculate $\int_{i}^{f} (L_1 - L_2) dt$ where $L_1 = f(t)$ and $L_2 = f(t+dt)$? Thanks. [on hold]

I have two functions f(u) and g(u). I have to use these two functions to calculate a function $h(u) = \int_{i}^{f} (L_1 - L_2) du$ where $L_1 = func(f(u),g(u))$ and $L_2 = func(f(u+du),g(u+du))$.
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0answers
34 views

What is the closed form of $ \int_0^1\int_0^1\frac{x^2 y^3}{\cos(\frac{\pi}{2} xy)}.dxdy $ [on hold]

What is the closed form of $ \int_0^1\int_0^1\frac{x^2 y^3}{\cos(\frac{\pi}{2} xy)}.dxdy $ This problem proposed by issa Khaled
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0answers
42 views

Possible wrong inequality.

Let $(A,\mathcal{F},\mu)$ be a measure space and $f$ a measurable real function. Define $\text{ess} \sup (f)=\inf\{c\in \mathbb{R}:\mu(\mid f\mid>c)=0\}$ and $\text{ess} \inf (f)=\sup\{c\in \mathbb{...
2
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1answer
99 views

Prove that shock wave is weak solution of Burgers' equation (Riemann problem)

In math modeling studies, I need to prove that $$u(x,t)=\begin{cases}u_l\qquad x<st\\ u_r\qquad x>st\end{cases}$$ where $$s=(u_l+u_r)/2$$ is a weak solution for the Riemann problem of ...
10
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2answers
234 views

Integral involving hypergeometric function $\int_0^1[{}_2F_1(\frac13,\frac23;1;x^3)]^2dx$

Question: How to prove $$I=\int_0^1\bigg[{}_2F_1\left(\frac13,\frac23;1;x^3\right)\bigg]^2dx=\frac{\sqrt3}{32\pi^5}\Gamma\left(\frac13\right)^9?$$ Source: An integral competition post of my country. ...
6
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4answers
213 views

Double Integral $\int\limits_0^a\int\limits_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac32}$

How to solve this integral? $$\int_0^a\!\!\!\int_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac32}$$ my attempt $$ \int_0^a\!\!\!\int_0^a\frac{dx \, dy}{(x^2+y^2+a^2)^\frac{3}{2}}= \int_0^a\!\!\!\int_0^a\...
36
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3answers
2k views

Integral $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful (yes beautiful) and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\...
1
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1answer
43 views

Solving gaussian integral $\int_1^3 e^{-\frac{t^2}{2}} \,dt$ with polar coordinates

Let's say I have the integral $\displaystyle \int_1^3 e^{-\frac{t^2}{2}} dt$. Since $\frac{t^2}{2} = ({\frac{t}{\sqrt 2}})^2$ I could thus say $u = \frac{t}{\sqrt 2}$, $\frac{du}{dt} = \frac{1}{\sqrt ...
0
votes
2answers
681 views

what is the weight function in numerical integration?

The weight function when integrating f(x) is p(x) and looks like $\int{p(x)f(x)} dx$ If we're integrating f(x) why do we need to multiply some weighting before integrating?
0
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2answers
61 views

For which $~p,~q~$ does $\int_2^∞ x^p⋅(\ln x)^q dx$ converges? [duplicate]

For which $~p,~q~$ does $$\int_2^∞ x^p⋅(\ln x)^q dx$$ converges? I have no idea how to start. What should I do?? Edit : Thanks for help!!
4
votes
1answer
40 views

Is the integration of a gaussian function divided by polynomial possible?

I am trying to evaluate the following integral: \begin{equation} I=\int_{-\infty}^{\infty}\exp\left \{-\frac{(u-1)^2}{2\sigma^2}\right\}\frac{1}{(u-x)^2+y^2}\mathrm{d}u \end{equation} where $x,y\in\...
0
votes
1answer
28 views

Can the sum property of integrals be stated for indefinite integrals?

Some time ago I learned about the following property of integrals: If $ f $ and $ g $ are bounded, integrable functions on $\color{red}{[a, b]}$, then so is $f + g$ and $$ \displaystyle \int_\...
1
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1answer
223 views

Integrating 2-form and change of variables question

There's a nice quote online that says: "The whole point of differential forms is that integration of forms is designed to work consistently with the change of variables formula. Or another way ...