Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
76,846 questions
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Different ways to integrate $\int_0^\infty \sin\left((cx)^2\right)dx$ [duplicate]
I am looking different ways to integrate $\int_0^\infty \sin(cx)^2dx$. Here is my approach
We know $$\int_0^\infty e^{-kx^2}dx=\frac{\sqrt\pi}{2\sqrt k}$$
Now if we substitute $k$ as $k^2$ and $k=\...
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One of the most difficult questions from the university entrance exam in Vietnam [closed]
A tortilla with a diameter of $20$ cm is placed inside a cylindrical mold with radius $4$ cm and folded along its diameter to form a taco shell.
Find the volume (in cubic centimeters) of the space ...
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Marginal integral being finite everywhere
Let $f(x,y)\geq 0$ be a continuous function on $[0,\infty)^2 \setminus \{0\}$. Suppose that for all $A \subseteq [0,\infty)^2 \setminus \{0\}$ that is bounded away from zero, we have that $\int_A f(x,...
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integrate $\int_{c}^{\infty} 2\sqrt{A t} K_1(2\sqrt{A t}) \exp{\left(-\lambda t\right)} dt $, where $A, \lambda, c$ are positive and greater than 0?
I want to integrate $I_1$. I start the evaluation by dividing it into two integral forms, $I_{11}$ and $I_{12}$. I solved $I_{11}$, but I find it difficult to solve $I_{12}$. Please help me to solve $...
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Integration by parts on definite integral
I have an integral,
$$
I = \int_a^b x f(x) dx
$$
and I would like to express this in terms of $\int_a^b f(x) dx$ if possible, but I don't see how integration by parts will help here. If $u = x$ and $...
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Evaluation of $ \int_0^\infty x^{2n}\ln\left(1+2e^{-x}\cos\alpha+e^{-2x}\right)\mathrm{d}x$
I am trying to obtain a closed-form solution for the following integral:
$$ I_n(\alpha) = \int_0^\infty x^{2n}\ln\left(1+2e^{-x}\cos\alpha+e^{-2x}\right)\mathrm{d}x $$
Here, $n$ is a non-negative ...
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Integral of the quantile function of a standard normal distribution
Let $\Phi$ denote the CDF of a standard normal distribution and $\phi=\Phi'$ its density. The claim is that
\begin{align*}
\int_{\alpha}^1 \Phi^{-1}(l)\ dl = \int_{\Phi^{-1}(\alpha)}^\infty l\phi(l)\ ...
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Why does the Darboux/Riemann integral exist for all partitions whose mesh goes to $0$ when it exists for only one? [closed]
I know that, if
$$\forall \epsilon >0, \exists P \;\text{ such that }\; U(f,P)-L(f,P)<\epsilon,$$ then $f$ is Riemann/Darboux integrable. But from that result, how can it be proven that $f$ is ...
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Existence of the Adjoint of a General Nonlinear Differential Operator [duplicate]
Let's call $\hat{D}$ some general differential operator which acts on a space of functions defined over a $d$ dimensional space (i.e. each function takes as input a set of input variables $\vec{x}$ ...
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Can $\int^\infty_{1+2\sqrt{3}}\frac{du}{\sqrt{u^3-15u-22}}$ be rewritten as the gamma functions etc.?
I am studying about elliptic integrals and elliptic curves.
I came across the following integral:
\begin{equation}I:=\int^{+\infty}_{1+2\sqrt{3}}\frac{du}{\sqrt{u^3-15u-22}}\tag{1.1}\end{equation}
I ...
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identification of bilinear inverse problem [closed]
Consider that $$h^z(t) = \int_0^t\int_0^t\gamma_1(t-\tau)\gamma_2(t-u)f^z(\tau,u)d \tau d u,$$ where $h^z(t)$ and $f^z(\tau,u)$ are known for all $z \in{1,\ldots,K}$ and $t\in[0,T]$, also assume that $...
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Evaluating $\int_{\pi/2}^{\pi} \frac{(\cos x - \sin x) ~dx}{\sin x + \cos x + \sec x + \csc x + \tan x + \cot x} $
In the light of a valid comment by @Ted Shifrin, I propose
to change the domain of the integral from $[0,\pi]$ to $[\pi/2,\pi]$
Here is a definite integral having all six elementary trigonometric ...
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Closed formula or bounds of area-to-volume ratio of spectral norm balls
In the matrix space $\mathbb{R}^{m\times n}$, define the spectral norm ball $$B_\lambda := \left\{ A \in \mathbb{R}^{m \times n} : \sigma_1(A) \leq \lambda \right\}$$ where $\sigma_1 (A)$ is the ...
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Closed form for an oscillatory Bose–Einstein-type integral?
Does the following integral admit a closed form in terms of standard constants and/or classical special functions
(e.g. Gamma, zeta, polylogarithms, exponential integrals, Fresnel/Airy functions, ...
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The asymptotic behavior of $\int_{-\infty}^\infty\frac{I_m'(|k|R)}{K_m'(|k|R)}e^{i(kz+m\theta)}K_m(|k|r_0)K_m(|k|r)dk$ as $R\to\infty$ where $r,r_0>R$
I would like to determine the asymptotic behavior of
$$
f_m (r,z; r_0,R) = \int_{-\infty}^\infty \frac{I_m'(|k|R)}{K_m'(|k|R)} \, e^{i(kz+m\theta)} K_m(|k|r_0) K_m(|k|r) \, dk \, ,
$$
in the limit $...
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