# P(cos(x)^3)/(x^6+1) very hard

1. Feb 1, 2012

### joao_pimentel

Hi guys

I'm wondering how shall I solve this primitive/integral

$$\int\frac{\cos^3 x}{x^6+1}dx$$

I really don't know how to start... which substitution shall I take or if I shall solve it by parts...

Any hints or tips will be very appreciated

2. Feb 1, 2012

3. Feb 1, 2012

### joao_pimentel

I don't know if I need an explicit expression, I just need to find how shall I start...

I've tried all trigonometric substitutions and nothing :(

I'm lost

4. Feb 1, 2012

### chiro

Hey joao_pimentel.

Have you looked at different tables of integrals? Chances are you will find a table out there somewhere that gives a form that is exact or 'close' to the form you are working with.

One strategy I have for you is to reduce the trig terms to something that is at least linear. You might have to do a few integrations by parts by differentiating the trig terms and then using trig conversions to put them in the appropriate form.

The denominator of the term looks like a (1 + x^2) which is used in many tables, although if its in combination with a trig term then that could still make it complicated.

5. Feb 1, 2012

### chiro

Also having looked at the solution, maybe another suggestion is to consider a transformation to an expontential form (in the form of using expressions of e^(whatever)) so for example cos(x) = 1/2(e^(ix) - e^(ix)).

6. Feb 1, 2012

### JJacquelin

Split the integral in two, in writting cos(x)^3 in terms of cos(x) and cos(3x)
Split both integrals in six, in writting 1/((x^6)+1) in terms of a sum of six terms c/(x-r) where each r is on the form r=(-1)^(1/6) , i.e. the six complex roots of unit.
Then, the antiderivatives of each cos(x)/(x-r) and cos(3x)/(x-r) can be formally expressed in terms of the special functions Ci(z) and Si(z) where z is a complex linear function of x.

7. Feb 1, 2012

### D H

Staff Emeritus
That is the factorization of x6-1, not x6+1. The latter factorization is a bit uglier.

First factor $x^6+1$ as $(x^2+1)(x^4-x^2+1)$. The latter term in turn has the factorization $(x^2-1+\sqrt[3]{-1})(x^2-\sqrt[3]{-1})$, so

$$x^6+1 = (x^2+1)(x^2-1+\sqrt[3]{-1})(x^2-\sqrt[3]{-1})$$

I'll leave the last ugly bit as an exercise for the OP.

8. Feb 1, 2012

### joao_pimentel

Thanks a lot guys... it was a big help...

I shall write cos(x)^3 in terms of a sum of cos(x) and cos(3x) as JJacquelin said and then fatorize x^6+1 as D H said... Then I shall use Ci(z) e Si(z) functions for each term..

I'll try it :)

Thanks a lot again

João

9. Feb 2, 2012

### joao_pimentel

Guys

I know that $$cos^3(x)=\frac{1}{4}\left(cos(3x)+3cos(x)\right)$$

Then

\begin{aligned} \int &\frac{\cos^3 x} {(x^2+1) (x^2-1+\sqrt[3]{-1}) (x^2-\sqrt[3]{-1})} dx \\ & =\frac{1}{4} \int \frac{\cos{3x}} {(x^2+1) (x^2-1+\sqrt[3]{-1}) (x^2-\sqrt[3]{-1})} dx + \frac{3}{4} \int \frac {\cos{x}} {(x^2+1)(x^2-1+\sqrt[3]{-1} )(x^2-\sqrt[3]{-1})} dx \end{aligned}

Then I need to factorize $$\frac{1}{(x^2+1)(x^2-1+\sqrt[3]{-1})(x^2-\sqrt[3]{-1})}$$ in partial fractions

$$\frac{1} {(x^2+1) (x^2-1+\sqrt[3]{-1})(x^2-\sqrt[3]{-1})} = \frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2-1+\sqrt[3]{-1})} +\frac{Ex+F}{x^2-\sqrt[3]{-1})}$$

Then I get each fraction $$\cos(a x)\frac{\alpha x + \beta}{x^2+\lambda}$$

How do I use such Ci(z) functions... Actually I confess I'm not aware of such Ci(z) and Si(z) functions... would it be Hyperbolic function???

Thank you a lot

Last edited by a moderator: Feb 2, 2012
10. Feb 2, 2012

### phyzguy

11. Feb 2, 2012

### JJacquelin

I think that it not a good idea to process with partial fractons which denominators are second degree polynomials. This will lead to complicated integrals, not easy to expess in terms of Ci and Si functions.
I maintain what I already said in my previous post [ Y, 01:19 PM] :
1/((x^6)+1) is much easier to write as the sum of six terms on the form c/(x-r) where each c is a different complex number easy to find and where the six different r are the roots of (r^6)+1=0, i.e. six complex number easy to find.

12. Feb 2, 2012

### D H

Staff Emeritus
Here's what you said yesterday:
That's incorrect. Using those (x-r) as the factors, where each r is one of the six sixth roots of unit, is the factorization of x^6-1=0, not x^6+1=0.

13. Feb 3, 2012

### JJacquelin

Obviously :
r = exp(i(2n+1)pi/6)
I think that it isn't the key point of the question.