# Tagged Questions

Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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### Convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$

Problem: Analyze the convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$. It seems to me that 'I' converges for $0<a<1$. My work: I wrote integral 'I' as a ...
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### Comparison test with improper integral

I have the integral $$\int_2^\infty\frac{3}{\sqrt[3]x(x+2\sqrt x)}dx$$ and have to find out whether it's divergent or convergent using the comparison test. I've been trying to understand this topic ...
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### Inner product on $\mathbb{R}[X]$

Let $P$ and $Q$ be two polynomials in $\mathbb{R}[X]$ and let $$\langle P,Q\rangle =\int _{-\infty}^{+\infty}P(x)Q(x)f(x)dx$$ with $f(x) = \frac{1}{\sqrt{2\pi}} \exp(-x^2/2)$. I would like to ...
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### Short-time Fourier Transform identity in $L^2$

Define the Short-time (or windowed) Fourier Transform of a function $f:\mathbb{R}\rightarrow\mathbb{C}$ as follows, $F_gf(\omega,t)=\int\limits_{\mathbb{R}}f(x)\overline{g(x-t)e^{ix\omega}}dx$. Show ...
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### Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$

I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
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### How much can the integrability at zero tell about the decay rate around zero?

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
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### $\int_{- \infty}^{+ \infty} |f(t)| dt < \infty \implies \int_{-\infty}^{x} f(t) dt$ is continuous?

I've found counter example for $(A),(D)$ and have shown except a bounded interval $F$ is uniformly continuous everywhere else. And so $(B)$ would imply $(C)$ is correct. But I can't show $(B)$ is ...
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### Juantheron-like integral

While seeing this post, the following integral is just struck me $$\int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1$$ I have tried like what user @OlivierOloa did in ...
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### Evaluate improper integral: Exponent of square root.

I was working in a problem in physics, which gave me this integral, and I need solution: $$\int_{-\infty}^\infty \exp{\left(-\sqrt{x^2 + a^2}\right)}dx$$ The problem is, I have no clue how to start....
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The problem is i need to study the convergence of A and B and find the antiderivative of C $$A=\int_0^\infty \frac{\sin(x) +x}{\sqrt x + x^3}dx$$ $$B=\int_0^\infty \frac{1}{\sqrt {e^x-1}(x^2+x^{1/... 3answers 74 views ### Trying to solve improper integral I've been trying to solve this$$ \int_{-\infty}^\infty {\sin(x)\over x+1-i }dx $$using residue theorem. I've tried using a square contour pi, pi+pii, -pi+pii, pi and half a circle but with the ... 1answer 98 views ### For what values of a does \int_0^\infty\left(\frac{x^a}{1 + x^2}\right)^4 \, dx converge? I'm learning about convergence/divergence of improper integrals and need help with the following problem: Find for what values of a does the following integrals exists$$(1) \int_0^\infty\...
"Using the substitution $t=\tan \frac{x}{2}$, prove that for every $-1<r<1$, $\int_{0}^{\pi}\frac{\cos x}{1-2r\cos x+r^2}dx=\int_{0}^{\pi}\frac{r}{1-2r\cos x+r^2}dx$ " I've tried the suggestion,...