# Tagged Questions

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### Evaluate improper integrals

I can't get an answer for these two questions... help? $\int_{-2}^0 \frac{x^2}{\left(1+x^3\right) ^2} dx$ $\int_{0}^1 \frac{\ln x}{x^\frac13} dx$ For question 1 I tried integrating but it gets ...
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### Is it possible? improper integral no answer?

Is it possible to evaluate this improper integral? If not then how should I answer?? $$\int_{0}^1 \frac1x dx$$
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### Mysterious inconsistency in an inhomogeneous linear 2nd order ODE with specified boundary values

I am trying to solve $$J''(\tau) = h (J - B) \tag{1}$$ where $h$ is a positive real constant and $B$ is a real-valued, smooth and otherwise well-behaved function for $\tau \in (0, \infty)$. I have ...
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### Problem related to the integration

Recently I read that $$\int_{-p^{2}}^{+p^{2}} \frac{1}{\sqrt{x^{2}-p^{2}}}dx$$ tends to a finite real number as $p \to 0$. Can anyone explain me why this is true?
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### Is it always possible to find a $t>0$, such that $\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|dx<C~~~?$

Is it always possible to find a $t>0$, such that $$\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|\,dx<C~~~?$$ where $C$ is independent of $n$. Here is my idea: We know that \begin{align} ...
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### $\lim_{\varepsilon \to 0}\int_{|x|>\varepsilon}\frac{e^{-(t-x)^2}}{x}dx=?$

I'm computing this integral $$\lim_{\varepsilon \to 0}\int_{|x|>\varepsilon}\frac{e^{-(t-x)^2}}{x}dx=?$$ I'm not sure that its integral whether exist. How could I solve it? Thanks for ...
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### Integral from $0$ to $\infty$ of $\frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right)$

Evaluate the integral $$\int_0^\infty \left( \frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right) \right) dx$$ I have read about ...
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### Limit of a singular integral

Denote $f_{\gamma}(x) =\frac { (1+\gamma)}{2} |x|^{\gamma}$. We consider: $$I(\gamma) = \int_{-1}^1\int_{-1}^1 \ln (|x-y|) f_{\gamma}(x) f_{\gamma}(y) dx dy$$ I would like to know the limit of ...
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### if the improper integral $\int^\infty_a f(x)\,dx$ converges, then $\lim_{x→∞}f(x)=0$ [closed]

I need to prove that: $$\lim_{x→∞}f(x)=0$$ if $$\displaystyle∫^∞_af(x)\,dx$$ converges. I need a proof or an specific, and if possible simple, counterexample. Would really appreciate your help! ...
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### Limit of the integral, integral of the limit (no dominated convergence)

I'm working on the integral: $\lim_{x \rightarrow 0} \int_0^T \frac{1}{\sqrt{\pi t}} \exp \left(-\frac{x^2}{t}-t\right) \left(1-\frac{t}{T} \right) \mbox{d}t$ I'ld like to inverse the integral and ...
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### Limit $\lim \limits_{z\rightarrow 0^+} z \int_1^\infty x^2 e^{-zx^2-zx} \mathrm dx$

Consider the following limit: $$\lim \limits_{z\rightarrow 0^+} z \int_1^\infty x^2 e^{-zx^2-zx} dx$$ Can we find the answer to this limit without calculating the integral? I'll be thankful if you ...
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Consider a continuous function $f:[0, \infty)\rightarrow\mathbb R$ such that $\lim_{x\rightarrow\infty} f(x)=1$. I would like to prove $$\lim_{n\rightarrow\infty}\frac{1}{n!}\int_0^\infty ... 1answer 122 views ### Commuting an \int improper at its both ends and \lim I am working on the following problem: Let f, g be continuous nonnegative functions defined and improperly-integrable on (0, \infty) Furthermore, assume they satisfy$$ \lim_{x\rightarrow ...
Say, we are interested in deriving $$\int_{-\infty}^{\infty}e^{-x^2}=\sqrt{\pi}\tag{1}$$ There are many well known ways to do it, for example: by polar coordinates via the gamma function, etc. ...