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 Jun23 asked A linear differential equation Jun23 comment Fourier transformation of $x^2 e^{-\lambda x}$ @Mercy as I mentioned it only exists in distributional sense. For normal functions the integral obviously diverges. Jun23 comment Fourier transformation of $x^2 e^{-\lambda x}$ In the distributional sense, using differentiation under the integral sign you'll get sth like $\delta''(\epsilon + i \lambda)$. Jun23 comment Sum inequality: $\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$ Possibly related: math.stackexchange.com/questions/13490/… Jun22 comment Prove that: $\int_{0}^{1} \frac{x^{4}\log x}{x^2-1}\le \frac{1}{8}$ Practically this solution uses @PeterTamaroff work, without it you could as well write right away $\frac{\pi^2}{8} - \frac{10}{9} \le \frac{1}{8}$ :/ Jun22 revised Prove that: $\int_{0}^{1} \frac{x^{4}\log x}{x^2-1}\le \frac{1}{8}$ added 209 characters in body Jun22 revised Prove that: $\int_{0}^{1} \frac{x^{4}\log x}{x^2-1}\le \frac{1}{8}$ edited body Jun22 answered Prove that: $\int_{0}^{1} \frac{x^{4}\log x}{x^2-1}\le \frac{1}{8}$ Jun22 answered Computing the derivative from the definition Jun22 answered Integral as a limit of a sum Jun22 answered How to calculate first variation of length of curve? Jun22 answered Prove that: $\frac1{20}\le \int_{1}^{\sqrt 2} \frac{\ln x}{\ln^2x+1} dx$ Jun21 comment $\int f(x) dx$ is appearing as $\int dx f(x)$. Why? @AppliedImagination actually I study physics :) Jun21 comment $\int f(x) dx$ is appearing as $\int dx f(x)$. Why? The argument I heard and which is quite convincing is that in physics $f(x)$ can have very long form. So in order not to forget about $dx$ we write it first :) Jun21 comment Limit of sum with binomial coefficient One way is to consider integral: $$\frac{1}{2i} \int_\gamma \sqrt{ \frac{\Gamma (n+1)}{\Gamma (n+1 - z) \Gamma (z+)}} \frac{1}{\sin \pi z} \, dz$$ where $\gamma$ is a rectangle with corners in $-\frac{1}{2} \pm ib$ and $n+\frac{1}{2} \pm ib$ than take limit $n, b \to +\infty$ which requires some manipulations along the road. Jun21 comment Aysmptotic relation @anon sorry, I meant $F$ and $G$. Jun21 comment Aysmptotic relation @anon we can't use de l'Hospital rule if we don't know limits of $f$ and $g$ when $x \to +\infty$. Jun20 comment Integration by means of complex analysis @PeterTamaroff oh, pardon me. Nice job :) Jun20 comment Integration by means of complex analysis @PeterTamaroff you still use P.V. for cosine which simply can't work. Jun20 comment Integration by means of complex analysis @PeterTamaroff I've got some link for you math.stackexchange.com/questions/9402/… btw today I discovered that here is something like FAQ for topics :D