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.
50,075
questions
0
votes
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 ...
0
votes
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 ...
1
vote
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'...
0
votes
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
votes
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 $$\left(\int_0^a x\cos ...
0
votes
0answers
19 views
Laplace transform to solve fredholm eqn
$$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
votes
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 ...
1
vote
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 $$I(n) = \operatorname{Li}_n^2(1)-2\...
0
votes
0answers
14 views
Multivariable function Integrable for what values?
For what values of $\alpha,\beta \in \mathbb{R}$ will the function
$$f:(0,1]\times(0,1] \to \mathbb{R}: (x,y) \mapsto x^\alpha y^\alpha (x+y)^\beta$$
be integrable?
Normally, I don't have problems ...
0
votes
0answers
57 views
Implication of finiteness of integral
Consider the measure space $(A,\mathcal{F},\mu)$. We say that a real measurable function $f$ on $A$ is integrable if $\int_A \mid f\mid d\mu < \infty$.
Furthermore, an integral of a real ...
11
votes
2answers
330 views
+100
Challenging integral: Evaluate $\int_0^1\frac{\ln^3(1-x)\operatorname{Li}_3(x)}{x}\ dx$
How to evaluate $$I=\int_0^1\frac{\ln^3(1-x)\operatorname{Li}_3(x)}{x}\ dx\ ?$$
I came across this integral $I$ while I was trying to compute two advanced sums of weight 7. The problem with my ...
2
votes
0answers
31 views
Does $\int_V f(s)ds=\int_U f(\varphi (t))|\det \varphi '(t))|dt$ hold if $f$ is measurable?
Let $\varphi :U\to V$ a diffeomorphism where $U,V$ are open. If $f:\mathbb R\to \mathbb R$ is a Borel function integrable function, we have that $$\int_V f(s)ds=\int_U f(\varphi (t))|\det \varphi '(t)|...
1
vote
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+\...
1
vote
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 ...
1
vote
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.
$$\mathbb{V}[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^...
1
vote
0answers
18 views
Saddle point method for a stationary subspace of dimension $> 0$
I recently asked a question on Physics SE regarding the validity of using the saddle point technique when the saddle point is not only degenerate, but forms a continuous subspace of the parameter ...
4
votes
0answers
279 views
How can one show the multiple integral $\int_{0}^{a} \int_{0}^{a} \frac{dx ~ dy}{(a^2+x^2+y^2)^{3/2}}= \frac{\pi}{6a}$ by hand? [duplicate]
Mathematica gives the value of the multiple integral
$$
\int_{0}^{a} \int_{0}^{a} \frac{dx ~ dy}{(a^2+x^2+y^2)^{3/2}}=\frac{\pi}{6a}.
$$
See also this result from WolframAlpha:
The question is ...
13
votes
3answers
1k views
Methods to evaluate $ \int _{a }^{b }\!{\frac {\ln \left( tx + u \right) }{m{x}^{2}+nx +p}}{dx} $
Today I saw a question with an answer that made me rethink of the following question, since it's not the first time I try to find an answer to it. If you look at the answer of Mhenni Benghorbal
here ...
0
votes
2answers
37 views
Why are some improper integrals convergent and others divergent?
The integral of the function $f(x)=1/x^2$ is convergent and it equals 1 when the limits of the integral is $\int_1^\infty$ but it's divergent and equals $\infty$ when the limits are $\int_0^1$.
I know ...
0
votes
0answers
13 views
Fubini's theorem for Borel functions with respect to Lebesgue measure
Is the following version of Fubini's theorem true? All the following integrals are with respect to Lebesgue measure.
Suppose $f(x,y):\mathbf{R}^n \times \mathbf{R}^m=\mathbf{R}^{n+m}\to \mathbf{R}\...
2
votes
1answer
41 views
Calculate $\int\sqrt{(x^2+1)}dx$
Calculate $I$ =$\int\sqrt{(x^2+1)}dx$
I have tried calculating it using integration by parts:
$$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\...
-4
votes
0answers
47 views
What is the closed form of $ \int_0^\infty \frac{x-\tanh(x)}{x^3} .dx $ [on hold]
What is the closed form of $ \int_0^\infty \frac{x-\tanh(x)}{x^3} .dx $
1
vote
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
$$\int_0^{\infty } \frac{\sinh (a x) \sinh (b x)}{(\cosh (a x)+\cos (t))^2} \, dx=\frac{\pi b \csc (t) \csc \left(\frac{\pi b}{a}\right) \sin \left(\...
1
vote
1answer
45 views
How to integrate $\exp(-|xy| - \phi(x^2+y^2))$?
I am trying to integrate this function:
$$
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp\{ -\lvert xy \rvert - \phi(x^2 + y^2) \} dx dy
$$
A little bit of background: the negative logarithm of ...
1
vote
1answer
53 views
How should I setup the integral to calculate surface area
Find the surface area of the part of the surface $y^2 +z^2 =2z$ cutoff by the cone
$x^2 = y^2+z^2$
So the given surface is actually a circle with radius $1$ and centre $(0,1)$ in y-z
plane,but I am ...
0
votes
1answer
24 views
Determine limit using DCT
I want to determine the following limit (for $\alpha>0$)
$$
\lim_{\alpha \to +\infty} \int_0^{+\infty} \frac{1}{\sqrt{x}(1+x^\alpha)}dx
$$
(call this integrand the fuction $f_\alpha$). Now, this ...
0
votes
0answers
28 views
Laplace Transform of e^{-at}
I have some trouble understanding the Laplace Transform of
$x(t) = e^{-at}u(t)$, where $u(t)$ is the Heaviside step function. When calculating the integral, we get to a point where
$$ X(s) = \int_{0}^...
13
votes
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 ...
0
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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 ...
-3
votes
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 $L_2 = func(f(u+du),g(u+du))$.
-3
votes
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
0
votes
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
votes
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
votes
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
votes
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
$$
\int_0^a\!\!\!\int_0^a\frac{dx \, dy}{(x^2+y^2+a^2)^\frac{3}{2}}=
\int_0^a\!\!\!\int_0^a\...
36
votes
3answers
2k views
Integral $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$
I am trying to calculate
$$
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
vote
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
votes
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
vote
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 ...
