# 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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### Line Integral, Work in physics

Hi there all: I have a problem! I need to find the work done on a particle that moves from $(0,0)$ to a point $(1,1)$ by a strait line $y=x$. The force acting upon the particle is $F = (y , 2x$). ...
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### Young inequality

I am trying to prove young's inequality for integrals $$ab \leq \int\nolimits_0^a \! f(x) \, \mathrm{d}x + \int_0^b \! f^{-1}(x) \, \mathrm{d}x.$$ Can you help me please?
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### Are there other analytic functions with this property of sinc function?

This question is motivated by my previous post about sinc function. Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere) function $f(x)$ on $\mathbb{R}$ ...
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### Integral of $\frac1{\cos x}$ using t substitution

Okay, so I'm trying to find $\int \frac1{\cos x}\mathrm{d}x$ using the substitution $t = \tan\left(\frac{x}{2}\right)$. I sub in the trig identity for $\sec$ as $\frac{1+t^2}{1-t^2}$ and then ...
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### Is the Riemann integral of a strictly smaller function strictly smaller?

We all know that if $f\leq{}g$ in $[a,b]$ then $$\int_a^bf\,dx\leq\int_a^bg\,dx$$ now, imagine that we have $f<g$, is it true that $$\int_a^bf\,dx<\int_a^bg\,dx$$
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### Why are there conflicting results for integration?

Using my calculator, I had tried to evaluate the definite integral $$\int_{0}^{1}x^x \mathbf{d}x$$ However, according to my CASIO $fx-570$ES, the result was Math ERROR, which led me to believe that ...
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### Limits of Integral $\lim\limits_{n\to\infty}\int\limits_a^b\sqrt[n]{f^n(x)+g^{n}(x)}dx.$

Let $f, g:[a,b]\to[0,\infty)$ be continuous functions. Find the value of $$\lim\limits_{n\to\infty}\int\limits_a^b\sqrt[n]{f^n(x)+g^{n}(x)}dx.$$ One said that the results is $\int\limits_a^b h(x)dx$ ...
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### Identity $\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$

I want to prove the identity $$F(z)=\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$$ First of all $F(z)$ defines an analytic function for $0<z<1$. I am little ...
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### Show that,$\int_0^\pi \left|\frac{\sin nx}{x}\right|\mathrm{d}x \ge \frac{2}{\pi}\left(1+\frac12+\cdots+\frac{1}{n}\right)$

Show that,$$\int_0^\pi \bigg|\dfrac{\sin nx}{x}\bigg|\mathrm{d}x \ge \dfrac{2}{\pi}\bigg(1+\dfrac12+\cdots+\dfrac{1}{n}\bigg)$$ I could not approach the problem at all. Please help.
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