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### Does the improper integral $\int_0^1 \frac{1}{x}\sin\frac{1}{x^2}$ exist?

Does the improper integral $\displaystyle\int_0^1 \frac{1}{x}\sin\frac{1}{x^2}$ exist? I don't know how to use the comparison test, and I cannot find a proper comparison function.
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### Evaluate $\int_{0}^{\infty} \frac{\sin{x}}{x}\,dx$ [duplicate]

how do I solve this integral ? $$\int_{0}^{\infty} \frac{\sin{x}}{x}\,dx$$ how to start ?
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### Parameterizing divots in contour integrals

I have recently attempted to pick up complex analysis and have been stuck on this problem for a few days: $$\int_{-\infty}^\infty \frac{\sin^2x}{x^2}\mathrm{d}x$$ Fortunately, it would seem that ...
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### Are there any ways of solving this intriguing integral? [duplicate]

I know a good number of methods to solve this integral like Feynman Trick, Laplace transform etc. $$\int_0^{\infty} \dfrac{\sin x}{x} \mathrm{d}x = \dfrac{π}{2}$$ Do y'all know some creative and ...
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### Why can't the indefinite integral $\int\frac{\sin(x)}{x}\mathrm dx$ be found? [duplicate]

I came across a list of functions in my calculus textbook whose indefinite integral cannot be found. It was written that the integral $$\int \frac{\sin(x)}{x} dx$$ cannot be evaluated without any ...
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### convergence dominate of Lebesgue-Application

my question is: let $f(x)= \dfrac{\sin (nx)}{x}.\varphi(x)$, where $\varphi \in C^\infty_c(\mathbb{R})$. We can found an function $g \in L^1(\mathbb{R})$ such as $|f(x)| \leq g$? Please. I think that, ...
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### Evaluating the integral: $\int\limits_{0}^{\infty}\left(\frac{\sin(ax)}{x}\right)^2 dx , a \neq 0$

Integrate from 0 to infinity $$\int\limits_{0}^{\infty}\left(\frac{\sin(ax)}{x}\right)^2 dx , a \neq 0$$ I tried evaluating the indefinite integral of that function using Sine integral. But I ...
### Confirm that $\int_{0}^{\infty}t^{-1}\sin t dt=\pi/2$ [duplicate]
Confirm that $\int_{0}^{\infty}t^{-1}\sin t dt=\pi/2$. The guide book I am using gives the following help: Consider $\int_{\gamma}z^{-1}e^{iz}dz$, where for $0<s<r<\infty$ the contour of ...