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23

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soft-question

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soft-question big-list

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linear-algebra matrices operator-theory factorial matrix-calculus

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integration soft-question definite-integrals

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Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

analysis integration definite-integrals closed-form zeta-functions

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110

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1

answer

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number-theory polynomials

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measure-theory summation

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sequences-and-series limits recurrence-relations

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sequences-and-series recreational-mathematics puzzle axiom-of-choice

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Why do we need to learn integration techniques?

calculus integration soft-question education

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combinatorics recreational-mathematics puzzle

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calculus real-analysis limits closed-form nested-radicals

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probability combinatorics graph-theory asymptotics expectation

May 12 '16 at 13:26 Community♦ 1

29

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Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

polynomials functional-equations contest-math

Oct 12 '15 at 16:54 Community♦ 1

26

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1

answer

695

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Prove that $ \sum\limits_{n=-\infty}^\infty\frac{\cos\pi\sqrt{n^2+1}}{3+4n^2}=\int\limits_{-\infty}^\infty\frac{\cos\pi\sqrt{x^2+1}}{3+4x^2}dx $?

calculus real-analysis integration sequences-and-series

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25

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4

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782

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Proof of $\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$

calculus real-analysis integration definite-integrals improper-integrals

Jan 25 '17 at 15:06 Community♦ 1

24

votes

4

answers

629

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Ways to show that $\int_{0}^{1}((1-x^r)^{1/r}-x)^ndx=\frac{1}{n+1}$

calculus integration definite-integrals

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2

answers

858

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Prove that the expression is a perfect square

algebra-precalculus elementary-number-theory functions

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456

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Does recursively replacing $\frac1n$ by $\frac1n(\frac12+\dots+\frac1{n+1})$ really converge to $\frac1e$?

real-analysis sequences-and-series convergence

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Suggestions for mathematical solitaire against boredom

soft-question recreational-mathematics

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Calculate moment of inertia of Koch snowflake

physics recreational-mathematics fractals fractal-analysis

May 19 '16 at 2:35 Bennett Gardiner 3,069

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Evaluate $\lim\limits_{x\to\infty}\frac1x\int_0^x\max\{\sin t,\sin(t\sqrt2)\}dt$

calculus integration definite-integrals

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297

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algebra-precalculus inequality recreational-mathematics absolute-value substitution

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5

answers

300

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Bounds (and range) of a nonlinear difference equation

recurrence-relations dynamical-systems periodic-functions

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16

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2

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555

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prove this $\sum_{n=1}^{\infty}\arctan{\left(\dfrac{1}{n^2+1}\right)}=\arctan{\left(\tan\left(\pi\sqrt{\dfrac{\sqrt{2}-1}{2}}\right)\cdots\right)}$

summation

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1

answer

192

views

Evaluating $\int_0^{\pi/2} \int_0^{\pi/2} \frac{\cos(x)}{ \cos(a \cos(x) \cos(y))} dx dy $

calculus real-analysis integration definite-integrals

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372

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combinatorics

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galois-theory radicals

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sequences-and-series

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Why don't we define “imaginary” numbers for every “impossibility”?

Surprising identities / equations

Really advanced techniques of integration (definite or indefinite)

How could we define the factorial of a matrix?

Some users are mind bogglingly skilled at integration. How did they get there?

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

The sum of an uncountable number of positive numbers

How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$

Predicting Real Numbers

Why do we need to learn integration techniques?

Proving a jigsaw is possible

What's the limit of $\sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + …}}}}}} $?

A disease spreading through a triangular population

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Prove that $ \sum\limits_{n=-\infty}^\infty\frac{\cos\pi\sqrt{n^2+1}}{3+4n^2}=\int\limits_{-\infty}^\infty\frac{\cos\pi\sqrt{x^2+1}}{3+4x^2}dx $?

Proof of $\int_0^\infty \frac{x^{\alpha}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi\sin (\alpha\beta)}{\sin (\alpha\pi)\sin \beta }$

Ways to show that $\int_{0}^{1}((1-x^r)^{1/r}-x)^ndx=\frac{1}{n+1}$

Prove that the expression is a perfect square

Does recursively replacing $\frac1n$ by $\frac1n(\frac12+\dots+\frac1{n+1})$ really converge to $\frac1e$?

Suggestions for mathematical solitaire against boredom

Calculate moment of inertia of Koch snowflake

Evaluate $\lim\limits_{x\to\infty}\frac1x\int_0^x\max\{\sin t,\sin(t\sqrt2)\}dt$

Quadratic inequality puzzle: Prove$ |cx^2 + bx + a| ≤ 2$ given $|ax^2+bx+c| ≤ 1$

Bounds (and range) of a nonlinear difference equation

prove this $\sum_{n=1}^{\infty}\arctan{\left(\dfrac{1}{n^2+1}\right)}=\arctan{\left(\tan\left(\pi\sqrt{\dfrac{\sqrt{2}-1}{2}}\right)\cdots\right)}$

Evaluating $\int_0^{\pi/2} \int_0^{\pi/2} \frac{\cos(x)}{ \cos(a \cos(x) \cos(y))} dx dy $

Finding the minimal number of members

On solvable octic trinomials like $x^8-5x-5=0$

Is $a_n=\sum_{\alpha=0}^n\;\text{round}(\sin\alpha)$ bounded?