Question

Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$

score 802 · Answer 1 · 2014-11-05 15:45:57

802

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Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$

Published on 05 Nov 2014 - 15:45

score 75 · Answer 2 · 2026-03-31 05:58:16

75

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score 96 · Answer 3 · 2026-03-28 20:03:26

96

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score 48 · Answer 4 · 2026-03-28 20:09:36

48

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Prove that $\int_0^\infty y^{\frac12} e^{-y^2} \int_0^\infty y^{-\frac12} e^{-y^2} =\frac{\pi}{2^{\frac32}}$

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score 142 · Answer 5 · 2014-11-05 20:20:15

142

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score 830 · Answer 6 · 2014-11-06 01:04:07

830

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score 5.9k · Answer 7 · 2014-11-06 11:56:48

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score 1.4k · Answer 8 · 2014-11-07 08:30:58

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score 1.8k · Answer 9 · 2026-03-31 05:41:40

1.8k

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score 56 · Answer 10 · 2014-11-07 23:25:35

56

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score 406 · Answer 11 · 2026-03-29 18:36:31

406

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Published on 29 Mar 2026 - 18:36

score 346 · Answer 12 · 2026-03-25 19:00:39

346

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score 892 · Answer 13 · 2014-11-09 16:52:15

892

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score 1.2k · Answer 14 · 2014-11-09 22:48:51

1.2k

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Published on 09 Nov 2014 - 22:48

score 308 · Answer 15 · 2014-11-09 23:32:35

308

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How to calculate $\int_0^\infty \frac{dx}{1+x^6}$

Published on 09 Nov 2014 - 23:32

List Question

Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$

Calculate this integral in $N$-dimensional space

Special Integral Proof

Prove that $\int_0^\infty y^{\frac12} e^{-y^2} \int_0^\infty y^{-\frac12} e^{-y^2} =\frac{\pi}{2^{\frac32}}$

Convergence of $\int_1^{+\infty}\frac{\sin(x)}{x}\arctan(x) \mathop{d}x$

Applying the Mean Value Theorem to improper Riemann integral(homework)

Positive to negative infinity integration

How to evaluate $\int _{-\infty }^{\infty }\!{\frac {\cos \left( x \right) }{{x}^{4}+1}}{dx}$

Find the leading order asymptotic behaviour of the integral

Integral with conditions

How to find closed-form of $\int_{0}^{+\infty} \operatorname{sech}^2 (x^2)\,dx$

Real integral giving a complex result

Evaluating a double integral from zero to infinity

Why is the improper integral $\int^0_{-\infty}{1\over 3-4x}dx$ divergent?

How to calculate $\int_0^\infty \frac{dx}{1+x^6}$

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