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
26 views
Cooling coffee question
I couldn't get the correct answer for this question:
A science geek brews tea at 195 ∘F, and observes that the temperature $T(t)$ of the tea after t minutes is changing at the rate of $$T'(t)=-6.5e^{-...
0
votes
1answer
54 views
Are there multiple ways to solve the integral $\int \cos^2(x)dx$
Both with the internet, places like wolfram and symbolab and in a previous question: How do you find singular solutions to first order differential equations?
I get different answers for the integral ...
1
vote
1answer
63 views
Integrate the Circumference of an Ellipse to Find the Area
For a circle of radius R, one can find the area by integrating the circumference equation in the interval $(0, R)$,
$$\text{Area} = \int^R_0 2\pi r\ dr = \pi R^2$$
My intuition for this is that we'...
0
votes
0answers
16 views
Laplace transform to solve fredholm eqn
$$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)-...
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 ...
4
votes
1answer
115 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 ...
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 $$\int_V f(s)ds=\int_U f(\varphi (t))|\det \varphi '(t)|...
1
vote
0answers
5 views
Derivation of Search Gradients for Natural Evolution Strategies
My question refers to chapter 2 of the Natural Evolution Strategies paper. I want to understand the derivation of the search gradients.
Given z, a solution vector sampled from a probability ...
1
vote
0answers
18 views
Saddle point method for a stationary subspace of dimension $> 0$
I recently asked a question on Physics SE regarding the validity of using the saddle point technique when the saddle point is not only degenerate, but forms a continuous subspace of the parameter ...
1
vote
1answer
41 views
Why is my derivation of the variance of a normal distribution incorrect?
I have seen a proof of the variance of a standard normal distribution, but am getting an incorrect answer am and not sure why. Here is my reasoning.
$$\mathbb{V}[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^...
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}\...
0
votes
2answers
36 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 ...
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:
$$f'(x) = 1, f(x) = x$$
$$g(x) = \sqrt{x^2+1}, g'(x) = \frac{x}{\sqrt{x^2+1}}$$
$$\int\sqrt{x^2+1}dx = x\...
-4
votes
0answers
47 views
What is the closed form of $ \int_0^\infty \frac{x-\tanh(x)}{x^3} .dx $ [on hold]
What is the closed form of $ \int_0^\infty \frac{x-\tanh(x)}{x^3} .dx $
1
vote
2answers
30 views
How to evaluate $\int_{1-\sqrt{1-t^2}}^{1+\sqrt{1-t^2}}(y^2-2y+t^2)dy$, where t is constant
I need to evaluate the following one. Can't understand the method in my textbook.
$$\int_{1-\sqrt{1-t^2}}^{1+\sqrt{1-t^2}}(y^2-2y+t^2)dy$$
My textbook is to let $\alpha=1-\sqrt{1-t^2}$, $\beta=1+\...
0
votes
1answer
24 views
Determine limit using DCT
I want to determine the following limit (for $\alpha>0$)
$$
\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}^...
1
vote
1answer
44 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
67 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 $$I(n) = \operatorname{Li}_n^2(1)-2\...
3
votes
2answers
51 views
Stopping Car Question
Suppose that as a yellow car brakes, its velocity is described by $$v(t)=3.3e^{1-t}-0.6$$
If the brakes are applied at time t=0 seconds, what is the distance it takes for the car to come to a ...
1
vote
1answer
62 views
Evaluate $\int_0^{\infty } \frac{\sinh (a x) \sinh (b x)}{(\cosh (a x)+\cos (t))^2} \, dx$
From Gradshteyn&Ryzhik 3.514.4 we know that
$$\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(\...
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 ...
-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{...
0
votes
0answers
56 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 ...
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
$$ \displaystyle \int_\...
4
votes
1answer
40 views
Is the integration of a gaussian function divided by polynomial possible?
I am trying to evaluate the following integral:
\begin{equation}
I=\int_{-\infty}^{\infty}\exp\left \{-\frac{(u-1)^2}{2\sigma^2}\right\}\frac{1}{(u-x)^2+y^2}\mathrm{d}u
\end{equation}
where $x,y\in\...
2
votes
1answer
58 views
Prove that $12(a\sin a+\cos a-1)^2\le 2a^4+a^3\sin(2a)$,$\forall a\in (0,\infty)$
Prove that $$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
1answer
64 views
Evaluating $\int^{\pi/2}_{0}x\cot (x)dx$ using Leibniz's integration rule
How does one evaluate the following improper integral using Leibniz's integration rule?
$$\int^{\frac{\pi}{2}}_{0}x\cot (x)dx$$
I tried to add a new parameter $\ln(\sec(tx))$
$$f(t) = \int^{\...
0
votes
1answer
18 views
For what values is the function integrable
For what values of $\alpha, \beta >0$ is the function
$$f: (0,+\infty) \to \mathbb{R} : x\mapsto \frac{1}{(x^\alpha + x^\beta)^2}$$
integrable?
Attempt
I believe there is no obstruction for the ...
-2
votes
1answer
66 views
Calculating $\int_{\pi/4}^{\pi/2} \frac{x \cos x-\sin x}{x} dx$ [on hold]
How to calculate this trigonometric integral ?
$$\int_{\pi/4}^{\pi/2} \frac{x \cos(x)-\sin(x)}{x} dx$$
1
vote
1answer
43 views
Solving gaussian integral $\int_1^3 e^{-\frac{t^2}{2}} \,dt$ with polar coordinates
Let's say I have the integral $\displaystyle \int_1^3 e^{-\frac{t^2}{2}} dt$.
Since $\frac{t^2}{2} = ({\frac{t}{\sqrt 2}})^2$ I could thus say $u = \frac{t}{\sqrt 2}$, $\frac{du}{dt} = \frac{1}{\sqrt ...
0
votes
0answers
19 views
Controllability Gramian via integration by parts
I'm trying to program something like this article. On page 3, there's the following equation (eq. 6) that should be expressible in closed-form:
$$\int_0^te^{(A(t-t'))} M e^{(A^T(t-t'))}dt'$$
where $...
2
votes
0answers
47 views
Let $f(x)$ is a stricly decreasing, continuous function from $[0, +\infty)$ to $[0, +\infty)$ such as $\lim_{x\to \infty} f(x)=0$ [duplicate]
Prove that $$\int_0^\infty {f(x)-f(x+1)\over f(x)}dx=\infty$$
I already understood that it is enough to prove that exist some constant $C$, such as for all $f$, exist $a_f$, such that $$\int_0^\infty ...
0
votes
0answers
31 views
How to handle derivatives inside the integral? [on hold]
Any ideas how to solve this?
I am a little bit unsure how to handle the derivatives inside the integral in this case.
$\int_{-\infty}^{\infty} \frac{d}{dx}(\phi i^3 \frac{d^3}{dx^3} \psi) dx$
0
votes
1answer
71 views
Difficult integrals with $\sin(\ln(1+x))$ [on hold]
Does anyone know how can I integrate these two functions :
$\dfrac{x}{(1+x)^{1.5}} * \sin (\ln (x+1))$
and
$\dfrac{(1-x)}{(1+x)^{1.5}} * \sin (\ln (x+1))$
Wolfram uses complex numbers and the ...
0
votes
2answers
42 views
Any idea how to integrate this over the reals without using wolfram alpha.
Any idea how to integrate this over the reals without using wolfram alpha?
Your welcome to go into $\Bbb{C}$ for the integration as long as its the same result over $\Bbb{R}$
I do recognize ...
0
votes
0answers
18 views
Approximating sum of Poisson distribution as an integral
given $P(n)=e^{-\mu}\frac{\mu^n}{n!}$, under what conditions can I meaningfully write
$$\sum_{n=0}^N P(n)\approx\int_0^N dx P(x) $$
using $x!=\Gamma(x+1)$? It is done nonchalantly in a document I'm ...
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 ...
8
votes
0answers
136 views
Evaluate $\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx$
From this page I found an interesting equality:$$\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx=\frac{1}{15} \left(\sqrt{2}+2+5 \log \left(\sqrt{2}+1\right)\right)$$...
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
0answers
28 views
Fourier transform of product of exponential decay and cumulative normal
I am trying to find the Fourier transform $ \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(t) \cdot e^{i\cdot\omega\cdot t} dt $ of the following function:
$$ f(t) = e^{-a\cdot t}\cdot \mathcal{N}\...
1
vote
2answers
52 views
All possible antiderivatives for Integration by Parts
Just had a quick inquiry with regards to the formula for Integration by Parts. If I'm not mistaken, the formula states that
$$\int f'(x)g(x) = f(x)g(x)- \int f(x)g'(x)$$
However, in the case that I ...
1
vote
1answer
51 views
Supremum of a function as the limit of an integral
I came across a problem and I really don't know where to start from. It states that:
$$\lim_{n\rightarrow\infty} \left(\int_a^b f(x)^n dx\right)^{1/n} = \sup \{ f(x) :a \leq x \leq b \}$$
with $f : [...
1
vote
0answers
24 views
Conditions for convergence of improper integral [duplicate]
I have come across the following problem in a calculus textbook:
We are asked to find the values of $ \beta \in \mathbb{R} $ for which the improper integral
$$ \int_{1}^{\infty}x^2 \cos({x^\...
1
vote
1answer
37 views
Nullity of a linear transformation, Tom Apostol calculus II problem
$Tom M Apostol, Calculus II$
$Exercise$ $ 2.4 $
$Q$ $26)$ Let V be a the linear space of all real functions continuous on [a,b]. If $ f$ $\in $ V, $g$=$T(f)$ means that
$ g(x)=\int_{a}^{b}\ f(t) ...
2
votes
1answer
48 views
Question about using an integral to calculate surface area and volume of a sphere.(dL instead of dx)
Why is it wrong to calculate the surface area of a sphere using $2\int_0^R 2πr dx$?
Why is $dA$ equals to $2πrdL$ and not $2πrdx$ ?
Why is $dS$ equals to $πr^2dx$ and not $πr^2dL$?
-2
votes
0answers
76 views
If $f(x)$ is integrable then $f(e^{x})$ is also integrable on $\Bbb R^{+}$ [on hold]
If $f(x)$ is integrable then is it true that $f(e^{x})$ is also integrable on $\Bbb R^{+}$?
1
vote
0answers
53 views
Is there any way to solve this integral? [on hold]
I'm having difficulty solving this integral, could someone help me?
$\int \frac {\tan \sqrt{x}}{x} dx $
Thanks in advance.
