# Tagged Questions

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 ...

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### If $f$ is continuously differentiable periodic, then $n\int_{0}^{1} f(x) \sin (2\pi nx) \mathrm dx \to 0$

If $f: \mathbb R \to \mathbb R$ is continuously differentiable periodic function of period $1$, then $$n\int_{0}^{1} f(x) \sin(2\pi nx)\mathrm dx \to 0$$ as $n\to\infty$.
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### Why can't Fubini's/Tonelli's theorem for non-negative functions extend to general functions?

Challenging-conventional-wisdom question based on an answer to my previous question. If $X \in L^1 (\Omega, \mathscr{F}, \mathbb{P})$ has pdf $f_X$, $Y \in L^1 (\Omega, \mathscr{F}, \mathbb{P})$ has ...
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### Finding $\overrightarrow v(t)$ and $\overrightarrow r(t)$ when given $\overrightarrow a(t)$

Suppose an object moves so that its acceleration is given by $\overrightarrow a(t) = \langle 0, -4 \cos(t), -3 \sin(t) \rangle$ with $\overrightarrow v(0)= \langle 0,0,3 \rangle$ and ...
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Let $T:\mathbb{R}^n\to\mathbb{R}^n$ be an isometry and let $f:D\to\mathbb{R}$ be a Riemann integrable function on $D$, i.e. such that the limit $$\int_D f(x_1,\ldots,x_n)\,d x_1\ldots ... 4answers 192 views ### How to integrate this fraction: \int\frac{1}{1-2x^2}dx? I'm not sure how to integrate this:$$\int\frac{1}{1-2x^2}dx$$I think it has to be this:$$ -2\cdot \arctan(x)$$Or this:$$\arctan(\sqrt{-2x^2})$$0answers 17 views ### Special name and the general solution method for the given integral I wonder whether the integral$$ I = \int \frac{d^3p}{(2\pi)^3}\frac{e^{i\vec{p} \cdot \vec{x} }}{\vec{p}^2+m^2}, $$has a special name or not and besides I am also interested in the general method ... 1answer 28 views ### Show f'(\beta) \int\limits_\beta^\alpha 1 dt \leq \int\limits_\beta^\alpha f'(t) dt \leq f'(\alpha) \int\limits_\beta^\alpha 1 dt Assuming f(x) is a function of single variable, and f'(x) is monotonically increasing Then claim: f'(\beta) \int\limits_\beta^\alpha 1 dt \leq \int\limits_\beta^\alpha f'(t) dt \leq f'(\alpha) ... 2answers 61 views ### Find the area of the region inside the inner loop of the​ limaçon Find the area of the region inside the inner loop of the​ limaçon r=7+14\cos(\theta)) So doing this problem, i got B the integral from 0 to 2pi (1/2)(7+14\cos(\theta))^2. and the Area as ... 1answer 19 views ### Finding the double integral when the boundaries for y is not specified A question in my Calculus book states: "Find the volume of the solid in the first octant bounded by the cylinder z=9-y^2 and the plane x=2" When this answer was being covered in class, it was ... 2answers 150 views ### Give 5 proofs for \int_0^{2 \pi} \ln(\frac{25}{16} - \sin(x)^2) dx = 0. When you ask my mentor : Am I any good at integrals ? You usually get an answer like this : Give 5 proofs for \int_0^{2 \pi} \ln( \frac{25}{16} - \sin(x)^2) dx = 0. I was able to show it with ... 2answers 126 views ### Evaluation of \int \sqrt{1+\cot x}dx? What is$$\int \sqrt{1+\cot x}dx$$My friends and I tried using all possible trigonometric formula. We couldn't find a way to solve it Please help me solve it. 0answers 46 views ### Convolution of exponential and rect functions I have a convolution question in my signals and systems problem set that is puzzling me:  f(t) = e^{-t/2T} u(t)  and  g(t) = rect(t/2T)  find the convolution f \ast g and I am assuming ... 1answer 63 views ### Completely missing the idea of this solution (finding an arbitrary function in integrand) I'm hoping that someone can explain this (partial) solution to me. In my textbook (Haberman PDEs book, q. 10.2.1), we're asked to find (complex) c(\omega) so that the following are equivalent (with ... 2answers 65 views ### Approximation for integral of Matrix Exponential I am trying to implement some algorithm in matlab. To this end, I need to discretise a system of differential equation as \dot x = A x + B u. Starting with initial condition x_0 the system after ... 2answers 125 views ### Prove that  \int_0^1 x^2 \psi(x) \, dx = \ln\left(\frac{A^2}{\sqrt{2\pi}} \right)  Basically what the title says: Prove that  \displaystyle \int_0^1 x^2 \psi(x) \, dx = \ln\left(\dfrac{A^2}{\sqrt{2\pi}} \right).  where A\approx 1.2824 denotes the Glaisher–Kinkelin ... 1answer 42 views ### Help with integration involving exponential I am trying to solve an equation in the book Digital Image Processing, but I am stuck in the steps in between the formula and solution. Here's the equation, the last line is the solution. Sorry it's ... 1answer 45 views ### Recursive formulas and integration [duplicate] Using integration by parts find a recursive formula of \int cos^n(x) dx and use it to find \int cos^5 x dx I have no idea how to do this and my knowledge does include integration by parts etc. I ... 0answers 17 views ### Domain in polar coordinates with a square and a discus I was doing some studying in Steward's Calculus when I came onto this problem. I am asked to integrate a certain function f(x,y) in this domain. I know how to do it when the inner boundary is a ... 5answers 178 views ### Integral \sqrt{1+\frac{1}{4x}}$$\mathbf\int\sqrt{1+\frac{1}{4x}} \, dx This integral came up while doing an arc length problem and out of curiosity I typed it into my TI 89 and got this output ...
Here's my work, without measure theory (Riemann-Lebesgue lemma) and without step function approximations: Let f(x) be continuous on the closed interval $[0,2\pi]$, and $x$ a real variable. Then we ...
Yesterday ago we learnt about surface integrals, and I already calculated some with parameters. One of these exercises was this one: $x= v\cos(u)$ $y= \sin(u)$ $z= v$ while \$B' = {(u,v): 0 \le ...