# Tagged Questions

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### Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
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### Counterexample for necessary condition of integrability

Can you give me an example of a non-negative function on $[0,1]$ that is NOT integrable, but $\lim_{t \to \infty} t \mu\{x : |f(x)| \geq t \} =0$?
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### Compute $\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$

Compute $$\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$$ According to Wolfram Alpha, the limit is zero. I tried to make substitution ...
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### numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
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### a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
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### Is integral convergent?

I have a problem with following integral: $\int_1^\infty \frac{\sqrt{x}}{1+x} \sin(2x)dx$ I was trying to prove convergence (or divergence) of this integral, however without any success. My best ...
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### Is the assumption $f \in C^4$ necessary for the composite Simpson's rule to be of order $p=4$?

In my introductory numerics class, we wanted to integrate a function $f \in C[a,b]$ numerically. After developing the Simpson's rule, we proved that if $f \in C^4$ then the composite Simpson's rule ...
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### How we compare the two following integral without calculation?

compare the two following integral without calculation : 1)$\displaystyle{\int_0^1x{e^{x^2}}dx}$ 2)$\displaystyle{\int_0^1 \sqrt{x}{e^{x}}dx}$ I would be interest for any comments or any replies
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### The integral on $[0,1]\times[0,1]$

Here I have a problem. $p$ and $q$ are positive numbers. the integral $$\int_0^1\int_0^1 \frac{1}{x^p+y^q}\;dx\;dy< \infty \Longleftrightarrow \frac{1}{p}+\frac{1}{q}>1$$ Here is my try, ...
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### convergence of a generalized Riemann integral

Could you please provide me some hints to test the convergence of this integral below? $$\int\limits_{0}^{+\infty}\dfrac{\sin x\cdot \sin 2x}{x^\alpha} \, dx$$ where $\alpha \in \mathbb{R}$
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### Convergence of $\displaystyle\int\frac{1}{\sqrt[3]{1-x^3}}\ dx$

Please help me to prove that this integral converges. $$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx$$ No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result ...
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### Question on regulated functions

Suppose that $f:[a,b]→\mathbb{R}$ is a regulated function. Then the integral $\int_a^bf(x)dx$ is deﬁned as the limit $\lim_{n→∞}\int_a^bs_ndx$, where $(s_n)_{n∈\mathbb{N}}$ is a sequence of step ...
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### Check the uniform convergence of parametric integral

While $0< \alpha <+ \infty$, prove if the parametric integral is uniform convergent on $\alpha$'s domain: $$\int^{+ \infty}_{0} e^{- \alpha x} \sin \beta x dx$$ $\beta$ is nonzero constant.
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### Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$\sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $X=(0,1)$? I was thinking about using the ...
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### Indefinite integral convergence

How can I prove that this integral converges? $$\int_0^1 \sqrt{\frac{1-kx^2}{1-x^2}} dx\quad\quad 0\leq k<1$$ Edit: fixed typo dt -> dx
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### Do simple functions converge almost everywhere?

Assume there is a sequence of simple functions s.t.: $$\|\int(s_m - s_n)\mathrm{d}\mu\|\to 0$$ Does it follow that there is a subsequence which converges almost everywhere? (Note the order of modulus ...
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### Multivariable Integral Calculus help

I have two questions. First: Is my proof "strong" enough? I am being asked to prove that $$\int_{0}^\infty\int_{0}^x e^{-sx}f(x-y,y) dydx = \int_{0}^\infty\int_{0}^\infty e^{-s(u+v)}f(u,v) dudv$$ ...
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### Check if $\int_1^{\infty\:}\left(e^{-\sqrt{x}}\right)dx$ converges using Convergence Test

I could use some help with an homework question: Using the convergence test, check if the following integral function converges or diverges (no need to calculate the limit itself): ...
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### A question about convergence of improper parametric integral

Could you give me some hint how to find all $\alpha\in R$ for with the integral $\int_0^1 \frac{a-x^{\alpha}}{1-x}$ converges. Is clear that this integral converges for all$\alpha\in N$, but I could ...
### Show that $\lim_{x\to\infty} \frac{1}{x} \int_0^x f(x) \ dx = \gamma$.
The Assignment Let $f: [0,\infty) \rightarrow \mathbb{R}$ be a function which is integrable on the intervall $[0,x] \ \forall x > 0$ and $\lim_{x\to\infty} f(x) = \gamma \in \mathbb{R}$. ...