For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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2answers
87 views
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Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$. So if I create a number line marking a-b and a+b. If that the integral above has 5 different answers depending on where (-t,t) is located on the number ...
10
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2answers
263 views
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A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
1
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1answer
85 views
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Is $\hat S$ a function?

Let $S(x)=\sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x)$. Find the Fourier transformation of $S(x)$. Is $\hat{S}$ a function? $$\hat{S}=\int_R \sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x) ...
1
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0answers
105 views
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Question on differentiate under integral

First we have the following theorem: Then we apply it to a concrete problem: Finally how to obtain the second rectangle?
8
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0answers
128 views
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Can we prove that the solutions of $\int_0^y \sin(\sin(x)) dx =1$ are irrational?

Can we prove that the solutions of $$\int_0^y \sin(\sin(x)) dx =1$$ are irrational? Wolfram Alpha gives two approximate sets of solutions as $\{4.58+2\pi k|k\in\mathbb{Z}\}$ and $\{1.69+2\pi ...