What is the solution for this integral?

[twoflower]

Hi,

I've been having troubles solving this integral:

[tex]
\int \frac{\sin x.\cos x}{\sin^4 x + x.\cos^4 x} dx
[/tex]

Here's how I tried it:

[tex]
t = \cos x
[/tex]

[tex]
dt = - \sin x dx
[/tex]

[tex]
dx = \frac{dt}{-\sin x}
[/tex]

[tex]
x = \arccos t
[/tex]

[tex]
dx = -\frac{1}{\sqrt{1-t^2}} dt
[/tex]

[tex]
-\frac{1}{\sin x} = -\frac{1}{\sqrt{1-t^2}}
[/tex]

[tex]
\sin x = \sqrt{1-t^2}
[/tex]

[tex]
\int \frac{\sin x.\cos x}{\sin^4 x + x.\cos^4 x} dx = - \int \frac{\cos x}{\sin^4 x + x.\cos^4 x} . \left( -\sin x \right) dx = - \int \frac{t}{\left(1-t^2\right)^2 + t^4.\arccos t} dt
[/tex]

Well, I don't know at all what to do with this now...

Does anybody have any idea?

PS: Why doesn't it insert newline, when I write "\\" in the LaTeX code?

 

[stunner5000pt]

in the [itex] cos^4 x [/itex] in the denominator how did you manage to sibstitute arcSin t into it
you know cos x dx = dt but [itex] cos^4 x [/itex] is NOT arccos t

be careful of your substitutions!

 

[dextercioby]

Is this your integral?

[tex] \int \frac{\sin x\cos x}{\sin^{4}x+x\cos^{4}x} \ dx [/tex]...?

If so,then i got good news and bad news

Bad:Maple & Mathematica can't crack it.
Good:You can forget about it.


Daniel.

 

[twoflower]

in the [itex] cos^4 x [/itex] in the denominator how did you manage to sibstitute arcSin t into it
you know cos x dx = dt but [itex] cos^4 x [/itex] is NOT arccos t

be careful of your substitutions!

I have no [itex]\arcsin x[/itex] nor [itex]\cos x dx = dt[/itex] there and I'm quite sure about the substitutions..

 

[twoflower]

Is this your integral?

[tex] \int \frac{\sin x\cos x}{\sin^{4}x+x\cos^{4}x} \ dx [/tex]...?

If so,then i got good news and bad news

Bad:Maple & Mathematica can't crack it.
Good:You can forget about it.

Daniel.

I know, I tried Maple and Mathematica too..So I have some doubts about whether the x in the denominator should be there..Anyway, if I ever met such an integral (with variables outside trigonometric functions), is there any way to solve it?

 

[dextercioby]

Very good,but that still doesn't help.

Daniel.

 

[dextercioby]

If it isn't there,the integral is easy.But in general this type of integrals involving "x" & transcendental functions are very seldom expressible in terms of "common" special functions.

Daniel.

 

[twoflower]

Very good,but that still doesn't help.

Daniel.

What did you reply with this to?

 

[stunner5000pt]

if there were no x in the denominator then it could be solved, it doesn't look as stubborn as the other integral

then [tex] \int \frac{\sin x\cos x}{\sin^{4}x+\cos^{4}x} \ dx = - \frac{1}{2} ArcTan (Cos(2x)) + C[/tex]

 

[twoflower]

If it isn't there,the integral is easy.But in general this type of integrals involving "x" & transcendental functions are very seldom expressible in terms of "common" special functions.

Daniel.

Yes, I tried it without the 'x' and it is easy. (BTW: Maple and Mathematica give different results. I got the Mathematica's one )

 

[twoflower]

if there were no x in the denominator then it could be solved, it doesn't look as stubborn as the other integral

then [tex] \int \frac{\sin x\cos x}{\sin^{4}x+\cos^{4}x} \ dx = - \frac{1}{2} ArcTan (Cos(2x)) + C[/tex]

Yes, that's the one I got too. Maple gives the same excepting the sign :)

 

[dextercioby]

Nope,Maple gives the same answer

[tex] -\frac{1}{2} \arctan\left(\cos 2x\right) +C [/tex]

Daniel.

 

[twoflower]

Nope,Maple gives the same answer

[tex] -\frac{1}{2} \arctan\left(\cos 2x\right) +C [/tex]

Daniel.

int( (sin(x)*cos(x)) / ( (sin(x))^4 + (cos(x))^4 ),x); [ENTER]
[tex]
1/2\,\arctan \left( -1+2\, \left( \cos \left( x \right) \right) ^{2}
\right)
[/tex]

 

[dextercioby]

My Maple is integrated in an ancient version of SWP.I guess it's less than 5.0 (i think,not too sure,though).So that's why it may be different from your answer.

I think your Maple is screwed up

Daniel.

 

[dextercioby]

Nope,the sign in the front of arctan is crucial.They're not the same function.

Daniel.

 

[twoflower]

My Maple is integrated in an ancient version of SWP.I guess it's less than 5.0 (i think,not too sure,though).So that's why it may be different from your answer.

I think your Maple is screwed up

Daniel.

I have Maple 9.01 ;) Strange.

 

[Spectre5]

haha, I just noticed the negative sign there and deleted my post :/

Didn't see it there before

 

[Spectre5]

hmmmmmm

I have Maple 9.52 and it gives me the correct answer (with the minus sign)