Question

Help with inverse trigonometric substitutions $ \int x^2\sqrt{a^2+x^2}\,dx $.

score 162 · Answer 1 · 2021-02-16 11:25:22

162

Views

Help with inverse trigonometric substitutions $ \int x^2\sqrt{a^2+x^2}\,dx $.

Published on 16 Feb 2021 - 11:25

score 61 · Answer 2 · 2021-02-17 03:10:27

61

Views

When $f$ is integrable, for what $g$ does $\int_{g(a)}^{g(b)} f(u) du = \int_a^b f(g(x))\cdot g'(x) dx$ hold?

Published on 17 Feb 2021 - 3:10

score 102 · Answer 3 · 2021-02-17 06:41:31

102

Views

showing $\int_{0}^{1}x\sqrt{1+4x^2}\,dx=\int_{1}^{2}\sqrt{(2-y)+\frac{1}{4}}\,dy$

Published on 17 Feb 2021 - 6:41

score 80 · Answer 4 · 2021-02-17 22:12:46

80

Views

Solve $\frac{dy}{dx}=\frac{y+y^2+x^2}{x}$

Published on 17 Feb 2021 - 22:12

score 54 · Answer 5 · 2021-02-18 13:07:44

54

Views

Substitution in ordinary differential equation gone wrong

Published on 18 Feb 2021 - 13:07

score 285 · Answer 6 · 2021-02-20 04:59:48

285

Views

Given two natural numbers $m, n$ so that $m> n.$ Prove that there exist any real numbers $x$ so that $2\sin nx\cos mx\geq 1$

Published on 20 Feb 2021 - 4:59

score 95 · Answer 7 · 2021-02-21 13:10:37

95

Views

Find the general solution to the differential equation $(x+1)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - y = (1+x)^2$

Published on 21 Feb 2021 - 13:10

score 104 · Answer 8 · 2021-02-21 23:36:47

104

Views

Factor $9x^4-37x^2+4$

Published on 21 Feb 2021 - 23:36

score 79 · Answer 9 · 2026-03-30 07:56:04

79

Views

Prove or disprove that $f\left(x\right)\leq 1,$ which is a continuous function so that $\int_0^x tf\left(2x-t\right){\rm d}t=\frac{\tan^{-1}x^2}{2}$

Published on 30 Mar 2026 - 7:56

score 110 · Answer 10 · 2026-03-27 01:04:16

110

Views

Question about Sui Zhen Lin's proof for inequality $\sqrt{\frac{a^2}{9b^2-8b+4}}+\sqrt{\frac{4b}{a+4}}\leq 1$ with positive numbers $a,b$ so $a+b=1$

Published on 27 Mar 2026 - 1:04

score 112 · Answer 11 · 2026-04-03 00:33:58

112

Views

Let $a$ be a non zero real number. Evaluate the integral $\int \frac{-7x}{x^{4}-a^{4}}dx$

Published on 03 Apr 2026 - 0:33

score 70 · Answer 12 · 2021-02-27 13:30:15

70

Views

Is it possible to eliminate a variable from these two differential equations?

Published on 27 Feb 2021 - 13:30

score 57 · Answer 13 · 2021-02-28 19:07:01

57

Views

A problem with Weirstrass substitution

Published on 28 Feb 2021 - 19:07

score 433 · Answer 14 · 2026-03-25 03:04:12

433

Views

Prove that there exist four polynomials $p_1,p_2,p_3,p_4$ in $x,y,z$ so that $(x^2+y^2+z^2)^3-8(z^3x^3+x^3y^3+y^3z^3)=p_1^2+p_2^2+p_3^2+p_4^2$

Published on 25 Mar 2026 - 3:04

score 692 · Answer 15 · 2026-02-23 20:39:32

692

Views

Prove that $\measuredangle\gamma= 90^{\circ}$

Published on 23 Feb 2026 - 20:39

List Question

Help with inverse trigonometric substitutions $ \int x^2\sqrt{a^2+x^2}\,dx $.

When $f$ is integrable, for what $g$ does $\int_{g(a)}^{g(b)} f(u) du = \int_a^b f(g(x))\cdot g'(x) dx$ hold?

showing $\int_{0}^{1}x\sqrt{1+4x^2}\,dx=\int_{1}^{2}\sqrt{(2-y)+\frac{1}{4}}\,dy$

Solve $\frac{dy}{dx}=\frac{y+y^2+x^2}{x}$

Substitution in ordinary differential equation gone wrong

Given two natural numbers $m, n$ so that $m> n.$ Prove that there exist any real numbers $x$ so that $2\sin nx\cos mx\geq 1$

Find the general solution to the differential equation $(x+1)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - y = (1+x)^2$

Factor $9x^4-37x^2+4$

Prove or disprove that $f\left(x\right)\leq 1,$ which is a continuous function so that $\int_0^x tf\left(2x-t\right){\rm d}t=\frac{\tan^{-1}x^2}{2}$

Question about Sui Zhen Lin's proof for inequality $\sqrt{\frac{a^2}{9b^2-8b+4}}+\sqrt{\frac{4b}{a+4}}\leq 1$ with positive numbers $a,b$ so $a+b=1$

Let $a$ be a non zero real number. Evaluate the integral $\int \frac{-7x}{x^{4}-a^{4}}dx$

Is it possible to eliminate a variable from these two differential equations?

A problem with Weirstrass substitution

Prove that there exist four polynomials $p_1,p_2,p_3,p_4$ in $x,y,z$ so that $(x^2+y^2+z^2)^3-8(z^3x^3+x^3y^3+y^3z^3)=p_1^2+p_2^2+p_3^2+p_4^2$

Prove that $\measuredangle\gamma= 90^{\circ}$

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