Intereting Posts

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Show that $d/dx (a^x) = a^x\ln a$.
Question about the proof that a countable union of countable sets is countable
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If a ring is Noetherian, then every subring is finitely generated?
How are simple groups the building blocks?
Unprovable statements in ZF
Say $X$ is $T_2$, $f: X \to Y$ is continuous, $D$ is dense in $X$ and $f|_D :D \to f(D)$ is a homeomorphism. Then $f(D) \cap f(X- D) = \emptyset$
Optimization of $2x+3y+z$ under the constraint $x^2+ y^2+ z^2= 1$
A question about the definition of $\mathbb{C}$
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conjectured general continued fraction for the quotient of gamma functions
(Locally) sym., homogenous spaces and space forms
Reference for trace/norm inequality
Given an infinite poset of a certain cardinality, does it contains always a chain or antichain of the same cardinality?

Calculate the integral

$$\int \ln (\sin x) \, dx.$$

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- Integral of the Von karman equation
- How to compute $\lim _{x\to 0}\frac{x\bigl(\sqrt{3e^x+e^{3x^2}}-2\bigr)}{4-(\cos x+1)^2}$?

Consider the following.

\begin{align}

I &= \int \ln(\sin(x)) \ dx

\end{align}

can be evaluated by integration by parts and leads to

\begin{align}

I &= x \ln(\sin(x)) – \int x \ \cot(x) \ dx \\

&= x \ln(\sin(x)) -x \ln(1 – e^{ix}) – \frac{i}{2} \left( x^2 + \operatorname{Li}_2(e^{2ix}) \right)

\end{align}

where $i =\sqrt{-1}$ and $\operatorname{Li}_2(z)$ is the dilogarithm function. It is of note that

\begin{align}

\int_0^{\pi/2} \ln(\sin(x)) \ dx = \frac{\pi}{2} \ln(2).

\end{align}

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- Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$
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