What is the solution for this integral?

• twoflower
In summary, the conversation discusses the difficulty of solving the integral \int \frac{\sin x \cos x}{\sin^4 x + x \cos^4 x} dx. The conversation includes different attempts at solving it using substitutions, but they all prove unsuccessful. It is mentioned that Maple and Mathematica are unable to solve it as well. Eventually, it is concluded that the integral is not easily expressible in terms of common special functions. A solution is provided for the integral without the variable 'x' in the denominator, but it is noted that this does not apply to the original integral. There is also some discussion about different versions of Maple giving different answers. In summary, the conversation discusses the difficulty of solving the integral \int
twoflower
Hi,

I've been having troubles solving this integral:

$$\int \frac{\sin x.\cos x}{\sin^4 x + x.\cos^4 x} dx$$

Here's how I tried it:

$$t = \cos x$$

$$dt = - \sin x dx$$

$$dx = \frac{dt}{-\sin x}$$

$$x = \arccos t$$

$$dx = -\frac{1}{\sqrt{1-t^2}} dt$$

$$-\frac{1}{\sin x} = -\frac{1}{\sqrt{1-t^2}}$$

$$\sin x = \sqrt{1-t^2}$$

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

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?

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

$$\int \frac{\sin x\cos x}{\sin^{4}x+x\cos^{4}x} \ dx$$...?

If so,then i got good news and bad news

Bad:Maple & Mathematica can't crack it.

Daniel.

stunner5000pt said:
in the $cos^4 x$ in the denominator how did you manage to sibstitute arcSin t into it
you know cos x dx = dt but $cos^4 x$ is NOT arccos t

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

dextercioby said:

$$\int \frac{\sin x\cos x}{\sin^{4}x+x\cos^{4}x} \ dx$$...?

If so,then i got good news and bad news

Bad:Maple & Mathematica can't crack 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?

Very good,but that still doesn't help.

Daniel.

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.

dextercioby said:
Very good,but that still doesn't help.

Daniel.

What did you reply with this to?

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

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

dextercioby said:
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 )

stunner5000pt said:
if there were no x in the denominator then it could be solved, it doesn't look as stubborn as the other integral

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

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

$$-\frac{1}{2} \arctan\left(\cos 2x\right) +C$$

Daniel.

dextercioby said:

$$-\frac{1}{2} \arctan\left(\cos 2x\right) +C$$

Daniel.

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

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.

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

Daniel.

dextercioby said:
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.

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

Didn't see it there before

hmmmmmm

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

Last edited:

1. What is an integral with sin() and cos()?

An integral with sin() and cos() is a type of mathematical function that involves integrating expressions containing sine and cosine functions. These functions are commonly used in physics, engineering, and other fields to describe periodic or oscillatory behavior.

2. How do you solve an integral with sin() and cos()?

The process for solving an integral with sin() and cos() is similar to solving any other integral. You can use integration techniques such as substitution, integration by parts, or trigonometric identities to simplify the expression and then evaluate the integral.

3. What is the difference between an integral with sin() and cos() and a regular integral?

The main difference is that an integral with sin() and cos() involves integrating expressions that contain trigonometric functions, while a regular integral can involve any type of mathematical function. This means that the methods used to solve these integrals may differ.

4. What are some common applications of integrals with sin() and cos()?

Integrals with sin() and cos() are commonly used in physics and engineering to model and solve problems involving periodic or oscillatory motion. They are also used in calculus to find areas under curves and to calculate volumes or other quantities.

5. How can I use integrals with sin() and cos() in real-life situations?

Integrals with sin() and cos() have many practical applications, such as calculating the displacement, velocity, and acceleration of objects in periodic motion. They are also used in fields such as signal processing, music theory, and computer graphics.

• Introductory Physics Homework Help
Replies
10
Views
372
• Introductory Physics Homework Help
Replies
16
Views
496
• Introductory Physics Homework Help
Replies
1
Views
226
• Introductory Physics Homework Help
Replies
6
Views
2K
• Introductory Physics Homework Help
Replies
28
Views
531
• Introductory Physics Homework Help
Replies
4
Views
307
• Introductory Physics Homework Help
Replies
7
Views
758
• Introductory Physics Homework Help
Replies
8
Views
694
• Introductory Physics Homework Help
Replies
19
Views
735
• Introductory Physics Homework Help
Replies
5
Views
687