Question

Integrate $\int_0^\pi\sqrt{\frac{1+\cos(2x)}{2}}dx$

score 283 · Answer 1 · 2026-04-03 01:13:32

283

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Integrate $\int_0^\pi\sqrt{\frac{1+\cos(2x)}{2}}dx$

Published on 03 Apr 2026 - 1:13

score 76 · Answer 2 · 2026-03-26 12:37:22

76

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Showing that $\int_0^{ \frac \pi 2} \cos^{2n+1}(x) dx \ = \frac{4^n(n!)²}{(2n+1)!}$?

Published on 26 Mar 2026 - 12:37

score 306 · Answer 3 · 2026-04-05 16:34:32

306

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Prove the integral inequality

Published on 05 Apr 2026 - 16:34

score 725 · Answer 4 · 2026-03-26 14:30:24

725

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Prove that: $e^{-\frac{|t|}{2}}= \int_{\mathbb R} \frac{e^{ixt}}{1+x^2} dx.$

Published on 26 Mar 2026 - 14:30

score 54 · Answer 5 · 2026-04-11 04:28:18

54

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Find multiple integral

Published on 11 Apr 2026 - 4:28

score 1.2k · Answer 6 · 2018-03-29 11:08:23

1.2k

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Why $\int\limits_{\sin(-5π/12)}^{\sin(5π/12)}\frac{dx}{1-x^2}=\ln\frac{1+\sin\frac{5π}{12}}{1-\sin\frac{5π}{12}}$?

Published on 29 Mar 2018 - 11:08

score 97 · Answer 7 · 2018-03-29 12:18:56

97

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On the integral $ \int \cos(x)^{{\cos(x)}}\mathrm{d}x$.

Published on 29 Mar 2018 - 12:18

score 75 · Answer 8 · 2026-03-25 01:38:27

75

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Can you explain the absolute value sign when integrating $\int_{-1}^{1}dx \int_{0}^{x^2} \sqrt{x^2-y} dy$?

Published on 25 Mar 2026 - 1:38

score 147 · Answer 9 · 2026-04-11 22:25:57

147

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integral involving hyperbolic tangent and cosine

Published on 11 Apr 2026 - 22:25

score 128 · Answer 10 · 2026-04-03 02:36:50

128

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Prove that $\int_0^\frac{\pi}{2}\cos^mx \sin^mxdx=2^{-m}\int_0^\frac{\pi}{2}\cos^mxdx$

Published on 03 Apr 2026 - 2:36

score 102 · Answer 11 · 2018-03-30 04:54:41

102

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Calculate $\int\limits_{0}^{1}{f\left( x \right)dx}$ when $f$ satisfies three given conditions.

Published on 30 Mar 2018 - 4:54

score 396 · Answer 12 · 2026-04-12 06:57:59

396

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Average value of $x\sin(x)$ from $0$ to $\infty$

Published on 12 Apr 2026 - 6:57

score 255 · Answer 13 · 2018-03-30 13:01:58

255

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Mistake in definite integral of $\int_{-1}^{1} d/dx \arctan(1/x) dx$

Published on 30 Mar 2018 - 13:01

score 40 · Answer 14 · 2026-04-11 06:51:35

40

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Complex integral over $\mathbb{R}$

Published on 11 Apr 2026 - 6:51

score 99 · Answer 15 · 2026-03-25 04:43:35

99

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Prove $\int_\pi^{2\pi}\frac{\left|\sin(nx)\right|}xdx\leq\frac2{\displaystyle\pi}(\frac1{n+1}+\frac1{n+2}+......+\frac1{2n})$

Published on 25 Mar 2026 - 4:43

List Question

Integrate $\int_0^\pi\sqrt{\frac{1+\cos(2x)}{2}}dx$

Showing that $\int_0^{ \frac \pi 2} \cos^{2n+1}(x) dx \ = \frac{4^n(n!)²}{(2n+1)!}$?

Prove the integral inequality

Prove that: $e^{-\frac{|t|}{2}}= \int_{\mathbb R} \frac{e^{ixt}}{1+x^2} dx.$

Find multiple integral

Why $\int\limits_{\sin(-5π/12)}^{\sin(5π/12)}\frac{dx}{1-x^2}=\ln\frac{1+\sin\frac{5π}{12}}{1-\sin\frac{5π}{12}}$?

On the integral $ \int \cos(x)^{{\cos(x)}}\mathrm{d}x$.

Can you explain the absolute value sign when integrating $\int_{-1}^{1}dx \int_{0}^{x^2} \sqrt{x^2-y} dy$?

integral involving hyperbolic tangent and cosine

Prove that $\int_0^\frac{\pi}{2}\cos^mx \sin^mxdx=2^{-m}\int_0^\frac{\pi}{2}\cos^mxdx$

Calculate $\int\limits_{0}^{1}{f\left( x \right)dx}$ when $f$ satisfies three given conditions.

Average value of $x\sin(x)$ from $0$ to $\infty$

Mistake in definite integral of $\int_{-1}^{1} d/dx \arctan(1/x) dx$

Complex integral over $\mathbb{R}$

Prove $\int_\pi^{2\pi}\frac{\left|\sin(nx)\right|}xdx\leq\frac2{\displaystyle\pi}(\frac1{n+1}+\frac1{n+2}+......+\frac1{2n})$

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