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3 questions
2
votes
4
answers
146
views
Solve the equation $\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}$
Solve in $\mathbb{R}$:
$
\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}
$
My approach:
Let $a = \sqrt{1-x}$ and $b = \sqrt{1+x}$ so $a^2 + b^2 = 2$. The ...
- 679
4
votes
1
answer
73
views
Proof of positivity of $~ x+\sqrt{x^2+1} ~$ of $~\operatorname{arsinh}(x)=\operatorname{arcsinh}(x)=\sinh^{-1}(x)= \ln \left( x+\sqrt{x^2+1}\right)$
Proof of positivity of $~ x+\sqrt{x^2+1} ~$
I found this formula appears at $~ \operatorname{arsinh}(x)= \ln \left( x+\sqrt{x^2+1}\right)~$
So, of course this argument inside the natural log function ...
- 1,539
0
votes
1
answer
117
views
Solve the equation $\sqrt{x^{2}+ 8}- \sqrt{x^{2}+ 3}+ 2x^{3}- x- 2= 0$ .
Solve the equation
$$\sqrt{x^{2}+ 8}- \sqrt{x^{2}+ 3}+ 2x^{3}- x- 2= 0$$
My solution is
$$x+ 2- 2x^{3}= \sqrt{x^{2}+ 8}- \sqrt{x^{2}+ 3}= \frac{5}{\sqrt{x^{2}+ 8}+ \sqrt{x^{2}+ 3}}\leq \frac{5}{\...
user552223