Mikhail Tikhonov
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Value of $\cos (z) $ given the value of $z $
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4 votes

Defenition : $\cos(z)= \frac{e^{iz}+e^{-iz}}{2}$ Evaluating: $$\cos(i\ln(2-\sqrt{3}))=\frac{e^{-\ln(2-\sqrt{3})} + e^{\ln(2-\sqrt{3})}}{2} = \frac{1}{4-2\sqrt{3}} + 1 -\frac{\sqrt{3}}{2} = 2$$ ...

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Calculus Integral $\int \frac{dt}{2^{t} + 4}$
3 votes

You've done the right move. Then since $$\frac{1}{u(u+4)}=\frac{1}{4u}-\frac{1}{4(u+4)}$$ You can get the answer $\dfrac{t \ln 2 - \ln (2^t+4)}{4 \ln2}$

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Finding radius of a circle inside of a circle.
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2 votes

Since there are n circles, angle $\theta = \frac{2\pi}{n}$(Complete angle should be $2\pi$). Let's now look at red triangle. Since it's isosceles, its median and angle bisector is the same line. So ...

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How would I integrate the following integral?
1 votes

It's possible to write down the antiderivative as some special function (in particular 2F1 hypergeometric function), but you don't need to evaluate the integral to prove it doesn't converge Hint: ...

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If $\{x_n\}$ is a sequence that diverges, then $\{cx_n\}$ diverges also for $c\ne 0$
1 votes

$\{x_n\}$ converges $\Leftrightarrow$ $\forall \epsilon > 0 \exists n_0 : \forall n,m > n_0 |x_m - x_n| < \epsilon$. Then $\{x_n\}$ diverges $\Leftrightarrow$ $\forall \epsilon > 0 \ \...

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Prove: If $x+y>22$ then $x>11$ and $ y>11$
1 votes

Ok, if $x<11$ AND $y<11$ you can sum it into $x+y<22$, but if its OR you have a complex, not a system, you can't sum it(even it would be eqs). For example, x=13 or y=2 doesn't mean x+y=15.

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A question about unitary block matrix
0 votes

I will try to complete the answer, mostly answering the follow-up question Rakesh Adhikesavan asked comments. Indeed if we consider $A,B,C,D\in M_n(\mathbb{R})$ with block matrix $P$ being $n\times n$ ...

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Is X a square in $\Bbb Z$?
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For $x^2 < 71, 50$ is not a square. So now we take $x^2 = 71 + 50$ and it turns out $11^2 = 121 = 50(\mod 71)$

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Limit of the Area of a function
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0 votes

Area of integration: $x\in[\lambda;0], \lambda\in(-1;0)$. Indefinite integral: $\int \frac{x^2 dx}{\sqrt{1+x^3}} = \frac{2}{3} \sqrt{x^3+1} + C$. Using FTC: $\int\limits_{\lambda}^{0} \frac{x^2 dx}{...

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Evaluate $\int \frac{\operatorname{csch}^2\sqrt u}{\sqrt{u}}$
0 votes

Hint: try $t=\sqrt{u}$ and $dt=\frac{du}{2\sqrt{u}}$

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Why is the absolute value of a equal to -a when a < 0?
0 votes

If $a<0$ you can consider it as $-|a|$ since $|a|$ is absolute value(like unsigned value). So $a=-|a|$, and $|a|=-a$ Example: $a=-1$, $|a|=1=-(-1)=-a$

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Interpreting equation of a plane
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0 votes

$x-2y+0z=0$ http://www.wolframalpha.com/input/?i=plane+x%3D2y $\vec{n}=\{1,-2,0\}$

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Switch to new variables
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0 votes

Ok, i will try to give my solution. Tell me if i'm right/wrong/unproofed. $$ \left\{ \begin{array}{c} \dot{x}x+t=0\\ 1+\dot{u}+\dot{y}=0\\ \end{array} \right. $$ Then since $y'=\frac{dy}{dx}=\frac{...

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$z(t) \mapsto \mathbb R$ or $z(t) \rightarrow \mathbb R$. Which notation is correct?
0 votes

$z(t\in[t_0;t_1])\in\mathbb{R}$ and to figure out that it's continuous, $z \in C^0(\mathbb{T})$, where $T=[t_0;t_1]$. If u want to use arrow it should be Dom(z) on left side.

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How to rotate the positions of a matrix by 90 degrees
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As best I understand the question, it's about how to transpose a matrix around its secondary diagonal. If I'm right, it might be something like this: $I^{T^{*}}=I_0A^TI_0$, where $(I_0)_{ij}=\delta_{n+...

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Find all integers n > 1 so that n, n + 2, n + 4 are all prime
0 votes

n=3; For other n check residues modulo 3

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