# Questions tagged [improper-integrals]

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 $\infty$ or $-\infty$, or as both endpoints approach limits.

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### Integrating a function composed with a Dirac delta [closed]

In a physics problem I found an integral of the form: $$\int d^4x\exp\left[i\delta(t-\tau)\hat{A}\right],$$ where $\hat{A}$ is an hermitian operator, $\tau\in\mathbb{R}$ and $d^4x=dtd^3x$. The ...
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### Estimation of a gamma function-like integral

A random variable $X$ has a pdf: $$f(x) = \frac{1}{k!} \cdot x^k \cdot e^{-x}$$ Prove that $$P(0<X<2\cdot(k+1)) > \frac{k}{k+1}$$ There are no conditions about $k$, so it can be non-integer. ...
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### For which values of the parameter $S$ is the integral convergent?

I feel like I should take lower limit and I did but I couldn't proceed. I need help. Should I use some comparison test 1 $$\int \limits^{2}_{1} \frac{(1- \cos \pi x)(x-1)^S}{x^3 -1} dx$$
1 vote
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### Laplace transform of $\sin(\omega t)$

I am learning about the Laplace transform and I know I got the answer to this example question wrong, but I'm trying to figure out if I just made a calculus or algebra type error, or if I'm ...
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### Can we convert the following integral equation to a differential equation:$h(r)= \int_0^\infty\frac{f(x)}{e^{r x} + 1} dx$?

Can we convert the following integral equation to a differential equation:$$h(r) =\int_0^\infty\frac{f(x)}{e^{r x} + 1 } dx?$$ Here, $f(x)$ is a non-trivial 'nice' function( whatsoever condition is ...
• 375
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### Is the Riemann Liouville fractional integral compact operator? [closed]

I am about to figured out that is the Riemann Liouville fractional integral compact operator or not? where f is continuous function in [0, b].
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### How do I find an equivalent for this integral?

I am trying to derive concentration bounds for the spectral norm of some rank-one matrices under Gaussian measure. My objective is to obtain a bound with respect to both the number of samples $N$ and ...
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### Are there examples of convergent improper integral but we can not apply FTC?

I want to find examples of improper integral such that (1) the lower and upper limits of the integral are finite (2) the improper integral is convergent, (3) the integrand is expressed as one equation,...
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1 vote
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### Decomposing a Fraction Involving Cube Roots for Integration

I had an exam the other day and there was this question to decide whether the following function is improperly integrable from 0 to 1. I wrote a solution for it but now I came to understand it's not ...
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### Show that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}$ is convergent.

Show that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}} \,dx$ is convergent. The points $0$ and $1$ are the only points of infinite discontinuities of $\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}.$ The integral ...
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### Integrals of asymptotic functions

If $f\sim g$ then these two integrals $$\int_a^{\infty}f(x)dx\text{ and } \int_a^{\infty}g(x)dx$$ are either both convergent or both divergent. Is this theorem correct? I have doubts because I think I ...
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### How to solve the integral $\int_0^\infty\frac{e^{-x} \sin x}{(e^{3 x} + 1) x^{3/10}} dx$

$$\mbox{How to solve the following integral ?}:\quad \int_{0}^{\infty}\frac{{\rm e}^{-x}\sin\left(x\right)}{\left({\rm e}^{3 x} + 1\right)x^{3/10}}{\rm d}x$$ I think it cannot be solved using ...
• 375
1 vote