Question

How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?

score 257 · Answer 1 · 2026-04-04 09:43:57

257

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How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?

Published on 04 Apr 2026 - 9:43

score 99 · Answer 2 · 2026-04-05 08:53:08

99

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Closed form of integration

Published on 05 Apr 2026 - 8:53

score 284 · Answer 3 · 2026-04-07 15:11:48

284

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Integral of ratio of polynomial

Published on 07 Apr 2026 - 15:11

score 311 · Answer 4 · 2026-03-30 13:24:10

311

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An inequality involving $\int_0^{\frac{\pi}{2}}\sqrt{\sin x}\:dx $

Published on 30 Mar 2026 - 13:24

score 65 · Answer 5 · 2026-04-04 21:57:16

65

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How is $\int_{-T_0/2}^{+T_0/2} \delta(t) \cos(n\omega_0 t)dt=1$ and $\int_{-T_0/2}^{+T_0/2} \delta(t) \sin(n\omega_0 t)=0$?

Published on 04 Apr 2026 - 21:57

score 81 · Answer 6 · 2026-04-08 01:35:10

81

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Roots of the quadratic eqn

Published on 08 Apr 2026 - 1:35

score 37 · Answer 7 · 2026-04-06 06:09:26

37

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Area between curves finding pressure

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score 162 · Answer 8 · 2026-03-29 14:00:07

162

Views

Hint required : Why is the integral $\int_0^x \frac{\sin(t)}{1+t}\mathrm{d}t$ positive?

Published on 29 Mar 2026 - 14:00

score 82 · Answer 9 · 2026-04-07 23:44:55

82

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A definite integral of a rational function: How can this be transformed from trivial to obvious by a change in viewpoint?

Published on 07 Apr 2026 - 23:44

score 131 · Answer 10 · 2026-04-02 06:34:40

131

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Integrate exponential over shifted square root

Published on 02 Apr 2026 - 6:34

score 255 · Answer 11 · 2026-04-01 05:49:06

255

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How to compute $\int_{0}^{1}\frac{x\arctan x}{1+(x+\frac{1}{x})\arctan x}dx$

Published on 01 Apr 2026 - 5:49

score 1.9k · Answer 12 · 2026-04-02 08:00:41

1.9k

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Compute $\int_3^6 \frac{\sqrt{3}}{x^2 - 6x +12} \,dx$

Published on 02 Apr 2026 - 8:00

score 117 · Answer 13 · 2026-04-02 05:41:06

$\int_0^{2\pi} \left(a\cos x+\sqrt{1-a^2\sin^2(x)}\right)^{2k} dt=2\pi\sum_{m=1}^k \binom{k}{m} \binom{k-1}{m-1}a^{2(k-m)}$

score 44 · Answer 14 · 2026-04-02 04:00:00

44

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Value of integration

Published on 02 Apr 2026 - 4:00

score 62 · Answer 15 · 2026-04-03 08:04:12

62

Views

Finding $\sum^{\infty}_{n=1}\int^{2(n+1)\pi}_{2n\pi}\frac{x\sin x+\cos x}{x^2}$

Published on 03 Apr 2026 - 8:04

List Question

How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?

Closed form of integration

Integral of ratio of polynomial

An inequality involving $\int_0^{\frac{\pi}{2}}\sqrt{\sin x}\:dx $

How is $\int_{-T_0/2}^{+T_0/2} \delta(t) \cos(n\omega_0 t)dt=1$ and $\int_{-T_0/2}^{+T_0/2} \delta(t) \sin(n\omega_0 t)=0$?

Roots of the quadratic eqn

Area between curves finding pressure

Hint required : Why is the integral $\int_0^x \frac{\sin(t)}{1+t}\mathrm{d}t$ positive?

A definite integral of a rational function: How can this be transformed from trivial to obvious by a change in viewpoint?

Integrate exponential over shifted square root

How to compute $\int_{0}^{1}\frac{x\arctan x}{1+(x+\frac{1}{x})\arctan x}dx$

Compute $\int_3^6 \frac{\sqrt{3}}{x^2 - 6x +12} \,dx$

$\int_0^{2\pi} \left(a\cos x+\sqrt{1-a^2\sin^2(x)}\right)^{2k} dt=2\pi\sum_{m=1}^k \binom{k}{m} \binom{k-1}{m-1}a^{2(k-m)}$

Value of integration

Finding $\sum^{\infty}_{n=1}\int^{2(n+1)\pi}_{2n\pi}\frac{x\sin x+\cos x}{x^2}$

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