Question

Prove that $f(ax + (1-a)y) = \frac{1}{y-x}\int_x^y f(t)dt$ implies $a = \frac{1}{2}$

score 485 · Answer 1 · 2018-02-04 12:19:16

485

Views

Prove that $f(ax + (1-a)y) = \frac{1}{y-x}\int_x^y f(t)dt$ implies $a = \frac{1}{2}$

Published on 04 Feb 2018 - 12:19

score 335 · Answer 2 · 2018-02-04 13:03:53

335

Views

Definite integral $\int _{-1}^1\:\frac{\arccos\left(x\right)}{1+x^2}dx$

Published on 04 Feb 2018 - 13:03

score 75 · Answer 3 · 2026-04-10 04:51:35

75

Views

Problem in finding the value of this limit

Published on 10 Apr 2026 - 4:51

score 235 · Answer 4 · 2018-02-04 14:33:12

235

Views

Computing $\int \sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\cdots}}}} \,\mathrm{d}x$

Published on 04 Feb 2018 - 14:33

score 78 · Answer 5 · 2026-03-27 11:48:53

78

Views

How to evaluate $\frac{1}{2\pi}\int \frac{1}{(1-\alpha \cos \omega)^2 +\alpha^2\sin^2 \omega } \, d\omega$

Published on 27 Mar 2026 - 11:48

score 83 · Answer 6 · 2018-02-04 17:55:31

83

Views

How to calculate $\int _\frac{3}{4}^{\frac{3}{2}} \sqrt{9 - 4x^2}$

Published on 04 Feb 2018 - 17:55

score 165 · Answer 7 · 2026-03-25 19:04:59

165

Views

Evaluate $\lim_{ n \to \infty} \sum_{r=2n+1}^{3n} \frac{n}{r^2-n^2}$

Published on 25 Mar 2026 - 19:04

score 475 · Answer 8 · 2026-04-09 00:33:37

475

Views

Evaluate $\int_0^1 \frac{\mathrm{d}x}{\sqrt{1-x^2}}\frac{x }{1-k^2x^2}\log\left(\frac{1-x}{1+x}\right)$

Published on 09 Apr 2026 - 0:33

score 385 · Answer 9 · 2026-03-25 07:16:34

385

Views

How to prove that the limit of an upper sum equals the integral of a function with these limited definitions?

Published on 25 Mar 2026 - 7:16

score 49 · Answer 10 · 2026-04-10 16:43:45

49

Views

Trigonometric exponential Integration

Published on 10 Apr 2026 - 16:43

score 2.6k · Answer 11 · 2026-04-10 16:57:51

2.6k

Views

How to find the fourier integral of this function?

Published on 10 Apr 2026 - 16:57

score 170 · Answer 12 · 2026-04-10 13:51:52

170

Views

Integration of the exponential radial function

Published on 10 Apr 2026 - 13:51

score 62 · Answer 13 · 2018-02-05 13:29:02

62

Views

Find the limit for the sum of a series with $a_k = \frac{1}{k\cdot 2^k}$

Published on 05 Feb 2018 - 13:29

score 45 · Answer 14 · 2026-04-10 03:26:52

45

Views

Finding relation between $A$ and $B$ is

Published on 10 Apr 2026 - 3:26

score 270 · Answer 15 · 2026-04-06 16:05:49

270

Views

Evaluate $\int_0^\infty\frac{\sin(\varphi_1x)}x\frac{\sin\varphi_2x}x\cdots\frac{\sin\varphi_nx}x\frac{\sin(ax)}x\cos(a_1x)\cdots\cos(a_mx) \, dx$

Published on 06 Apr 2026 - 16:05

List Question

Prove that $f(ax + (1-a)y) = \frac{1}{y-x}\int_x^y f(t)dt$ implies $a = \frac{1}{2}$

Definite integral $\int _{-1}^1\:\frac{\arccos\left(x\right)}{1+x^2}dx$

Problem in finding the value of this limit

Computing $\int \sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\cdots}}}} \,\mathrm{d}x$

How to evaluate $\frac{1}{2\pi}\int \frac{1}{(1-\alpha \cos \omega)^2 +\alpha^2\sin^2 \omega } \, d\omega$

How to calculate $\int _\frac{3}{4}^{\frac{3}{2}} \sqrt{9 - 4x^2}$

Evaluate $\lim_{ n \to \infty} \sum_{r=2n+1}^{3n} \frac{n}{r^2-n^2}$

Evaluate $\int_0^1 \frac{\mathrm{d}x}{\sqrt{1-x^2}}\frac{x }{1-k^2x^2}\log\left(\frac{1-x}{1+x}\right)$

How to prove that the limit of an upper sum equals the integral of a function with these limited definitions?

Trigonometric exponential Integration

How to find the fourier integral of this function?

Integration of the exponential radial function

Find the limit for the sum of a series with $a_k = \frac{1}{k\cdot 2^k}$

Finding relation between $A$ and $B$ is

Evaluate $\int_0^\infty\frac{\sin(\varphi_1x)}x\frac{\sin\varphi_2x}x\cdots\frac{\sin\varphi_nx}x\frac{\sin(ax)}x\cos(a_1x)\cdots\cos(a_mx) \, dx$

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