Question

Prove that $\frac{1}{1+a_1+a_1a_2}+\frac{1}{1+a_2+a_2a_3}+\cdots+\frac{1}{1+a_{n-1}+a_{n-1}a_n}+\frac{1}{1+a_n+a_na_1}>1.$

score 240 · Answer 1 · 2016-01-11 22:48:32

240

Views

Prove that $\frac{1}{1+a_1+a_1a_2}+\frac{1}{1+a_2+a_2a_3}+\cdots+\frac{1}{1+a_{n-1}+a_{n-1}a_n}+\frac{1}{1+a_n+a_na_1}>1.$

Published on 11 Jan 2016 - 22:48

score 191 · Answer 2 · 2016-01-12 18:59:55

191

Views

Beginner Integration (Substitution)

Published on 12 Jan 2016 - 18:59

score 87 · Answer 3 · 2016-01-14 21:28:24

87

Views

integral of a square root function by substitution.

Published on 14 Jan 2016 - 21:28

score 2.3k · Answer 4 · 2016-01-15 07:17:44

2.3k

Views

Prove that $ax+by+cz+2\sqrt{(xy+yz+xz)(ab+bc+ca)}\le{a+b+c}$

Published on 15 Jan 2016 - 7:17

score 246 · Answer 5 · 2016-01-16 09:30:26

246

Views

Integral with more-dimensional substitution variables

Published on 16 Jan 2016 - 9:30

score 196 · Answer 6 · 2016-01-18 03:49:09

196

Views

Prove that $ \sum\limits_{cyc} \sqrt{\cot{A}+\cot{B}} \ge 2\sqrt2$

Published on 18 Jan 2016 - 3:49

score 172 · Answer 7 · 2016-01-18 05:11:19

172

Views

Prove that $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}\geq1+4\sqrt{\frac{abcd}{(a+b)(b+c)(c+d)(d+a)}}$

Published on 18 Jan 2016 - 5:11

score 108 · Answer 8 · 2016-01-18 12:25:48

108

Views

Contour integral to real integral: find suitable change of variables

Published on 18 Jan 2016 - 12:25

score 870 · Answer 9 · 2026-03-29 12:24:17

870

Views

Solving integral using trig substitution $\tan(x/2)=t$

Published on 29 Mar 2026 - 12:24

score 1.9k · Answer 10 · 2016-01-18 21:31:57

1.9k

Views

Integration by substitution of $x\log(x)$

Published on 18 Jan 2016 - 21:31

score 172 · Answer 11 · 2016-01-18 22:49:26

172

Views

Sketch the domain, reverse the integral's limits and solve the double integral using the substitution method

Published on 18 Jan 2016 - 22:49

score 106 · Answer 12 · 2016-01-20 15:33:08

106

Views

Prove that the integral of $\frac{dx}{\sqrt{x-x^2}}$ is equal to $2\sin^{-1}\sqrt x+C$ using $u = \sqrt{x}$.

Published on 20 Jan 2016 - 15:33

score 469 · Answer 13 · 2016-01-20 15:57:36

469

Views

Prove that $2\sin^{-1}\sqrt x - \sin^{-1}(2x-1) = \frac{\pi}{2}$.

Published on 20 Jan 2016 - 15:57

score 1.4k · Answer 14 · 2016-01-23 23:40:09

1.4k

Views

If $m = 6x + 5$, what equation is equivalent to $(6x + 5)^2 - 10=-18x - 15$ in terms of $m$?

Published on 23 Jan 2016 - 23:40

score 52 · Answer 15 · 2026-03-30 15:14:08

52

Views

Double angle substitution in $\int_2^3 \sin x (\cos 2x)^4 \,\text{d}x$

Published on 30 Mar 2026 - 15:14

List Question

Prove that $\frac{1}{1+a_1+a_1a_2}+\frac{1}{1+a_2+a_2a_3}+\cdots+\frac{1}{1+a_{n-1}+a_{n-1}a_n}+\frac{1}{1+a_n+a_na_1}>1.$

Beginner Integration (Substitution)

integral of a square root function by substitution.

Prove that $ax+by+cz+2\sqrt{(xy+yz+xz)(ab+bc+ca)}\le{a+b+c}$

Integral with more-dimensional substitution variables

Prove that $ \sum\limits_{cyc} \sqrt{\cot{A}+\cot{B}} \ge 2\sqrt2$

Prove that $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}\geq1+4\sqrt{\frac{abcd}{(a+b)(b+c)(c+d)(d+a)}}$

Contour integral to real integral: find suitable change of variables

Solving integral using trig substitution $\tan(x/2)=t$

Integration by substitution of $x\log(x)$

Sketch the domain, reverse the integral's limits and solve the double integral using the substitution method

Prove that the integral of $\frac{dx}{\sqrt{x-x^2}}$ is equal to $2\sin^{-1}\sqrt x+C$ using $u = \sqrt{x}$.

Prove that $2\sin^{-1}\sqrt x - \sin^{-1}(2x-1) = \frac{\pi}{2}$.

If $m = 6x + 5$, what equation is equivalent to $(6x + 5)^2 - 10=-18x - 15$ in terms of $m$?

Double angle substitution in $\int_2^3 \sin x (\cos 2x)^4 \,\text{d}x$

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