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4

$begingroup$

Within this AoPS thread it is asked to evaluate the following integral

$$mathfrak I~=~int_0^infty fracxsin xcos x+cosh^2 xmathrm dxtag1$$

In order to be precise there is also a possible closed-form conjectured which is given by

$$mathfrak I~=~G-frac12tag2$$

But as it is pointed out within the linked thread this seems to be only a reasonable approximation off after the $5$th decimal digit.

I have to admit that it is highly improbable that there exists a nice looking closed-form for $(1)$ since the integrand involves polynomials, trigonometric aswell as hyperbolic functions. I am not even sure how to get started, i.e. which substitution to choose or which technique at all to start with.

A related, but perhaps more handable integral, would be the following

$$mathfrak J~=~int_0^infty fracsin xcos x+cosh^2 xmathrm dxtag1$'$$$

Out of experience I could imagine that $(1')$ may has a closed-form in terms of known constants $($or series$)$ since it only contains the two closely connected trigonometric and hyperbolic functions.

Is it in fact possible to deduce a closed-form for $(1)$ and $(1')$? For myself I cannot offer an approach since everything I tried was not helpful at all hence I was not even able to perform one or two steps in order to simplify the given integrals. I would be glad to see a full solution or even attempts in evaluating $(1)$ and $(1')$ since I have no idea how to deal with such integrands.

Thanks in advance!

EDIT

Out of pure chance I just stumbled upon a related MSE question dealing with the integral

$$int_0^inftyfracxsin^2(x)cosh(x)+cos(x)dx=1$$

Which on the other hand motivates me to believe that there may be a closed-form for $(1)$.

integration definite-integrals closed-form

share|cite|improve this question

edited Mar 19 at 19:47

mrtaurho

asked Mar 19 at 17:33

mrtaurhomrtaurho

6,09271641

$endgroup$

add a comment | 

4

$begingroup$

Within this AoPS thread it is asked to evaluate the following integral

$$mathfrak I~=~int_0^infty fracxsin xcos x+cosh^2 xmathrm dxtag1$$

In order to be precise there is also a possible closed-form conjectured which is given by

$$mathfrak I~=~G-frac12tag2$$

But as it is pointed out within the linked thread this seems to be only a reasonable approximation off after the $5$th decimal digit.

I have to admit that it is highly improbable that there exists a nice looking closed-form for $(1)$ since the integrand involves polynomials, trigonometric aswell as hyperbolic functions. I am not even sure how to get started, i.e. which substitution to choose or which technique at all to start with.

A related, but perhaps more handable integral, would be the following

$$mathfrak J~=~int_0^infty fracsin xcos x+cosh^2 xmathrm dxtag1$'$$$

Out of experience I could imagine that $(1')$ may has a closed-form in terms of known constants $($or series$)$ since it only contains the two closely connected trigonometric and hyperbolic functions.

Is it in fact possible to deduce a closed-form for $(1)$ and $(1')$? For myself I cannot offer an approach since everything I tried was not helpful at all hence I was not even able to perform one or two steps in order to simplify the given integrals. I would be glad to see a full solution or even attempts in evaluating $(1)$ and $(1')$ since I have no idea how to deal with such integrands.

Thanks in advance!

EDIT

Out of pure chance I just stumbled upon a related MSE question dealing with the integral

$$int_0^inftyfracxsin^2(x)cosh(x)+cos(x)dx=1$$

Which on the other hand motivates me to believe that there may be a closed-form for $(1)$.

integration definite-integrals closed-form

share|cite|improve this question

edited Mar 19 at 19:47

mrtaurho

asked Mar 19 at 17:33

mrtaurhomrtaurho

6,09271641

$endgroup$

add a comment | 

$begingroup$

Within this AoPS thread it is asked to evaluate the following integral

$$mathfrak I~=~int_0^infty fracxsin xcos x+cosh^2 xmathrm dxtag1$$

In order to be precise there is also a possible closed-form conjectured which is given by

$$mathfrak I~=~G-frac12tag2$$

But as it is pointed out within the linked thread this seems to be only a reasonable approximation off after the $5$th decimal digit.

I have to admit that it is highly improbable that there exists a nice looking closed-form for $(1)$ since the integrand involves polynomials, trigonometric aswell as hyperbolic functions. I am not even sure how to get started, i.e. which substitution to choose or which technique at all to start with.

A related, but perhaps more handable integral, would be the following

$$mathfrak J~=~int_0^infty fracsin xcos x+cosh^2 xmathrm dxtag1$'$$$

Out of experience I could imagine that $(1')$ may has a closed-form in terms of known constants $($or series$)$ since it only contains the two closely connected trigonometric and hyperbolic functions.

Is it in fact possible to deduce a closed-form for $(1)$ and $(1')$? For myself I cannot offer an approach since everything I tried was not helpful at all hence I was not even able to perform one or two steps in order to simplify the given integrals. I would be glad to see a full solution or even attempts in evaluating $(1)$ and $(1')$ since I have no idea how to deal with such integrands.

Thanks in advance!

EDIT

Out of pure chance I just stumbled upon a related MSE question dealing with the integral

$$int_0^inftyfracxsin^2(x)cosh(x)+cos(x)dx=1$$

Which on the other hand motivates me to believe that there may be a closed-form for $(1)$.

integration definite-integrals closed-form

share|cite|improve this question

edited Mar 19 at 19:47

mrtaurho

asked Mar 19 at 17:33

mrtaurhomrtaurho

6,09271641

$endgroup$

share|cite|improve this question

edited Mar 19 at 19:47

mrtaurho

asked Mar 19 at 17:33

mrtaurhomrtaurho

6,09271641

share|cite|improve this question

edited Mar 19 at 19:47

mrtaurho

asked Mar 19 at 17:33

mrtaurhomrtaurho

6,09271641

asked Mar 19 at 17:33

mrtaurhomrtaurho

6,09271641

asked Mar 19 at 17:33

mrtaurhomrtaurho

6,09271641