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MathFail's user avatar
MathFail
  • Member for 2 years, 6 months
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20 votes
Accepted

Integral of $\displaystyle \int\sqrt{x\sqrt{x\sqrt{x...\sqrt{x}}}}\,dx$ where $x$ is repeated $n$ times

19 votes
Accepted

How to prove $\frac{\sqrt{5}+1}{2}>\log_23$?

14 votes
Accepted

Why does limit: $\lim_{x\to\infty}\prod\limits^{\lfloor x\rfloor}_{n=1}\frac{\lceil x\rceil}{\lfloor x\rfloor}=e$?

13 votes
Accepted

How to compute: $\dfrac{2\cos 80^\circ-\sin 70^\circ}{\cos 70^\circ}$

13 votes
Accepted

Show that $\int_0^\pi \frac{x\cos x \sinh x}{\sin^2x+\sinh^2x}dx=\frac{\pi}{2}\ln\left(\frac{e^\pi+1}{e^\pi-1}\right)$

10 votes
Accepted

Calculating $\;\int_{-\infty}^{\infty}\frac{1}{(e^x-1-x)^2+4\pi^2}~dx$

9 votes
Accepted

Solve: $\dfrac{2}{x-6+\sqrt{x}}+\dfrac{3}{\sqrt{x}-2}=\dfrac{4}{\sqrt{x}+3}$

9 votes

Is there any simple method to evaluate $\int \frac{\tan (3 x)}{\tan(x)+\sec(x)} d x$?

8 votes
Accepted

Solve the equation: $ 3(\sin^3 x + \cos^3 x) + 4(\sin^7 x + \cos^7 x) = 7(\sin^5 x + \cos^5 x) $

8 votes

Evaluating $ \int \frac{ 1 } { \sin x - \cos x - 1} dx$

8 votes
Accepted

Evaluate series: $\sum_{k=1}^\infty (-1)^k\left[ k\ln\left(\frac{k^4+2k^3+k^2}{k^4+2k^3+3k^2+2k+2}\right)+\ln\frac{k^2+2k+1}{k^2+1} \right]$

8 votes
Accepted

Which number is bigger: $e^{\sqrt{5}} + e^{\sqrt{7}}$ or $2e^{\sqrt{6}}$?

8 votes

Solve trig equation: $\sin{(70^{\circ}-x)}=\frac{1}2 + \frac{1}{4\cos{(20^{\circ})}}$

8 votes

Convert integral $\int_0^{\frac{\sqrt3}2} {\frac{1}{{{x^{3/4}}{{\left( {1 + x} \right)}^{3/4}}}}dx}$ to elliptic integral

7 votes
Accepted

Can we solve $\frac{dx}{dt} = 3x+3t$ by "multiplying by $dt$"?

7 votes
Accepted

Prove $\int_0^{\pi/4} \frac{x \cos 2x}{1-\sin x} \mathrm dx > \frac{1}{4}$

7 votes

A Tough Series: $\sum_{n=0}^\infty2^na_{n+1}\sqrt{a_n}=\frac{\Gamma^2(\frac14)-4\cdot\Gamma^2(\frac34)}{8\sqrt{2\pi}}$

7 votes
Accepted

Solve the tower equation: $\displaystyle x^{x^{x^{x+1}+x+1}}=2$

7 votes

Maximal subgroup of a finite group contains all other subgroups.

7 votes

What is the simple way to solve the equation $x+\sqrt{a+\sqrt{x}}=a$ for all real $x$?

7 votes
Accepted

How to solve this ODE $~ y''+ (1-x^2)y = 0 $?

6 votes

Integral with closed form? $\int_0^\infty\frac{x^n\ln x}{(1+x)^2(1+x^m)}dx$

6 votes

Find the max/min value of $x^2+y^2$ subject to $3x^2+5xy+3y^2=1$

6 votes

Evaluate the limit: $\lim_{n\to\infty} n \sin (n! \cdot \pi \cdot e)$

6 votes

Find the value $\sum_{n=a}^b\frac1{\sin (2^{n+3})}$

6 votes
Accepted

A novice substitution in differentiation, why is it wrong?

6 votes

Evaluate : $\int_0^{\frac{\pi}{2}} \frac{\ln(\sec ^2 x)\sec^2x }{(\sec ^2 x+1)\tan x}\mathrm dx$

6 votes
Accepted

Compute the integral $\int_{0}^{\pi/2} \cos^a (x) \sin (a x) \ dx$

6 votes

How to solve $\sec^4(x)=\tan^4(x)+3\tan(x)$

6 votes
Accepted

Recursive formula for $\int_0^1 x^n (x + a)^{-1} \,dx$

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