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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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^{-...
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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 ...
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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'...
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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)-...
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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
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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 ...
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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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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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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 ...
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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]^...
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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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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 ...
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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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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 $
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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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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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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}^...
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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 ...
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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\...
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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 ...
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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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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 ...
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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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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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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{...
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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 ...
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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
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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\...
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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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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^{\...
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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 ...
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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 ? $$\int_{\pi/4}^{\pi/2} \frac{x \cos(x)-\sin(x)}{x} dx$$
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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 ...
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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 $...
2
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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 ...
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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. $\int_{-\infty}^{\infty} \frac{d}{dx}(\phi i^3 \frac{d^3}{dx^3} \psi) dx$
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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 ...
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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 ...
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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 ...
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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 ...
8
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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
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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 ...
0
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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}\...
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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
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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 : [...
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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^\...
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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
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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 $πr^2dL$?
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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 $\Bbb R^{+}$?
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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.