# Tagged Questions

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### The geometric interpretation [duplicate]

In the course of mathematical analysis, there was one problem that i excited to know more about it: What is the geometric interpretation of $$\int_a^b f(x)\,d(\alpha(x))$$ and $\alpha(x)$ is ...
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### A problematic integral: $\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$

Is there a special trick to calculate this integral? $$\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$$ for $\lambda>0$.
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### Show derivative of integral equals integral of partial derivative if M[0,1]-measurable

I am trying to determine a method of approaching the following: Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
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### Function that is Riemann-Stieltjes integrable but not Riemann integrable?

This is my first question, so please go easy on me :3 - I've searched, and I haven't found any questions that are particularly similar to this one. I'm reading Rudin's Principles of Mathematical ...
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### Highly Oscillating Integrals

I'd like to know the behavior of integrals of the form: $$\int_0^1 f(x) \cos(k x) dx$$ as $k \rightarrow \infty$ where f is a smooth function. It is easy to see, by expanding f in power series, ...
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### Computing an explicit solution to an integral equation via the Neumann Series.

I am hoping that someone can provide guidance for solving the integral equation $$u=f+\lambda Au$$ where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
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### Integral with hyperbolic functions

I need to compute: $$\int_{x^2+y^2=1} \frac{\sinh(x)dy- \sin(y)dx}{\cosh(x)-\cos(y)}$$ where the circle $x^2 + y^2 = 1$ is oriented anticlockwise. So, can somebody show me how? I found the ...
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### Calculate volume of inequality

$\{(x,y,z) \in \mathbb{R}^3 | 2*max(|x|,|y|)^2+z^2\leq 4\}$ any tips for me anyone? i made a sketch but what now?
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### Concept of integration to differential form

How to integrate differential form actually. As far as I know, a differential form is a multilinear function mapping from a vector space to a real number. Let's take $\int_c fdx+gdy$ as an example. It ...
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### For all integrable $f:[-1,1]\mapsto \mathbb{R}$ prove that $\int_{-1}^1f^2(x)\ge\frac12(\int_{-1}^1f(x))^2+\frac32(\int_{-1}^1xf(x))^2$

For all integrable $f:[-1.1]\mapsto \mathbb{R}$ peove that $$\int_{-1}^1f^2(x)dx\ge\frac12\left(\int_{-1}^1f(x)dx\right)^2+\frac32\left(\int_{-1}^1xf(x)dx\right)^2$$ Thanks in advance.
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### how prove this integral inequality?

How prove that for all continuous and decreasing function $f:[0 ,1]\mapsto(0,+\infty)$ $$\frac{\int_{0}^1x(f(x))^2dx}{\int_{0}^1xf(x)dx}\leq \frac{\int_{0}^1(f(x))^2dx}{\int_{0}^1f(x)dx}$$ thanks in ...
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### Showing that $\lim_{\delta \to 0^+} \frac{1}{\delta} \int_x^{x + \delta} f(t) \ \mathrm{d}t = f(x)$

I'm working on proving the following equation: $\lim_{\delta \to 0^+} \frac{1}{\delta} \int_x^{x + \delta} f(t) \ \mathrm{d}t = f(x)$, where $f$ is given to be Riemann integrable and continuous on ...
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### Proof of Integral and Limes

I need an idea for the following statement: $f: [a,b]\to\mathbb{R}$ continuous and $x\in (a,b)$ $\implies$ $\lim\limits_{h \rightarrow 0}{\frac{1}{2h} \int_{x-h}^{x+h} f(y) \, dy = f(x) }$ I just ...
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### Simplifying an integral by introducing an additional parameter.

This is one of my homework tasks this week. Calculate the integral $$I = \int_0^\infty dx\ x^3 e^{-x}$$ by introducing an additional parameter $\lambda$ and rewriting the exponential function as ...
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### Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.

Let $y(x)$ be a solution to $y''+e^xy=0$. Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.