# 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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### Area of the polar figure enclosed by the circle $r=2$ and the cardioid $r=2(1+cos θ)$

This is exercise 7, of the book Engineering Mathematics by Stroud, Chapter 24, Further Problems section. Here's a graph i made of the figure as i see it: It gives the answer as $π+8$. The integral ...
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### A limit in a Feynman “proof” about Fermat's Theorem.

As perhaps some of you already know, Richard P. Feynman, the famous physicist tried a non-orthodox (in his usual way, I suppose) proof of the Fermat's Last Theorem. He tried a probabilistic "proof" ...
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### On an inequality for harmonic functions

I'm trying to understand the following; $| u(z)|^{p}= |\int P_{z}u \ d\lambda |^{p} \le (\int P_{z} |u |^{p} \ d\lambda) (\int P_{z})^{p-1}$ where $P_z$ is the Poisson kernel and $u$ is harmonic in ...
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### Proving that an integral of several cdf and pdf functions is increasing in a certain parameter.

Basic assumptions: $n\geq3$, $a\leq b\leq c$, $b$ is simply a dummy variable of integration, and $\rho\geq0$. $F(z)$ and $f(z)$ represent the usual general CDF and PDF (no specified distribution here)....
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### Closed-form Solution for series involving incomplete Gamma Function

I am working on a solution for an intgeral that leads to a series that I am stuck at. Below is what I have done and how I got to the final series. Any ideas on how to solve the series at the end? \...
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### Approximation of series using integral

In notes of statistical physics I found the following approximation $$\sum\limits_{n=0}^{\infty}F\left(n+\frac{1}{2}\right)\approx \int_{0}^{\infty}F(x)dx+\frac{1}{24}F'(0)$$ for $F$ such that the ...
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### Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx$ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
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### How small need it be to approximate integral as one area of product of initial value times length.

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ How small need $\Delta t$ be to approximate $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$$ as $$a(t)\Delta t$$ [ Just one ...
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### Volumes by cylindrical shells

Find the volume bound by $$xy=1;x=0;y=1;y=3;$$rotated about $x$-axis. In my attempt, I determined that the shell height will be $1/y$ and the shell radius will be $y-1$ Clearly these are wrong as ...
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### What is the area between two first order taylor series approximations as they become closer to eachother

Let's say that $y=\sin{x}$. Then the first order taylor series approximation about $c$ is $g(x)=\sin{(c)}+\cos{(c)}(x-c)$. Note that this is also equivalent to the line tangent to the curve $\sin{x}$ ...
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### $\iiint xyz~ dx~dy~dz$ over positive octant sphere $x^2+y^2+z^2= 4$? Please help me about about question solution [on hold]

$\iiint xyz~ dx~dy~dz$ over positive octant sphere $x^2+y^2+z^2= 4$? I have problems figuring out the limits. Complete answer will be appreciated. Please help me about about question solution
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### Showing propagation in ordinary differential equation

I have a very simple linear First Order Homogeneous Differential Equation with time but this makes me ponder upon the definition of derivative and integral. $$\frac{dy}{dt} = -H(t)y$$ where $y, H(t)$ ...
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### Deriving the volume of a sphere through integation

Disclaimer: I'm basing my information off of this link in the case where I do not provide enough context. All calculus I've learned has been through curiosity, digging up old textbooks and google (I ...
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### Integral of a summation related to $\sin$ expansion

I am trying to evaluate the following integral. It has similarity to the Maclaurin expansion for $\sin$. $$\int_{-\infty}^\infty{\sum_{n=0}^{\infty}\frac{(-1)^n}{\left(2n+x^2\right)!}}\text{dx}$$ ...
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### Evaluating $\int\frac{dx}{(a\sin x+ b\cos x)^2}$, $a\neq 0.$

Could you just show the hint to solve this integral, please?
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### Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
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### Algebra $A$ and its Gelfand spectrum

Let $A$ be the set of all function $f$ on $\mathbb{R}$ of the form $$f(x)=d+\int\limits_{0}^{\infty}e^{ixt}k(t)dt,\qquad\quad x\in\mathbb{R},$$ where $d\in\mathbb{C}$ and $k\in L_1([0,\infty])$. The ...
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### Any tips to solve this integral : $I_1 = \int \ln(x^2)e^{\sin(x)}\sin(x^{\cos(x)}) dx$

Background: I was making new expressions to see whether I could efficiently find their derivatives... After having done that, I've started trying to integrate most of them; obviously most of them don'...
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### Integration by parts: is this legitimate way of using?

Is it legitimate to write $$\int_0^a\mathrm{d}x\,f(x)g(x)=\left[f(x)\int_0^x\mathrm{d}x\,g(x)\right]_0^a-\int_0^a\mathrm{d}x\,\frac{\mathrm{d}f(x)}{\mathrm{d}x}\int_0^x\mathrm{d}x\,g(x)$$ Thanks.
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