Integrate cos(x^5): Solve \int \cos x^5 dx

[Artaxerxes]

Can you solve this $\int \cos x^5 dx$ ?

 

[arildno]

No, I can't

 

[rocomath]

nope ...

 

[Zurtex]

It probably can't be done in terms of elementary functions. Mathematica has an answer of 2 different forms:

Take E(v,z) here to be this function http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/ we have:$$\int \cos x^5 dx = - \frac{1}{10} \left( E\left( \frac{4}{5} , -i x^5 \right) + E\left( \frac{4}{5} , i x^5 \right) \right)$$

Or in terms of the gamma function http://functions.wolfram.com/GammaBetaErf/Gamma/ :

$$\int \cos x^5 dx = - \frac{\left( x^{10} \right)^{\frac{4}{5}}}{10x^9}\left( \left(i x^5 \right)^{\frac{1}{5}} \Gamma \left( \frac{1}{5} , -i x^5 \right) + \left(-i x^5 \right)^{\frac{1}{5}} \Gamma \left( \frac{1}{5} , i x^5 \right) \right)$$

I don't really understand why mathematica didn't simplify it further, so I've tried to keep it to what mathematica outputted.

 

[Artaxerxes]

Thank you!

 

[DavidSmith]

thats why series solutions are so conviennt

 

[Gib Z]

The Taylor series for cos converges for all real x, so you should have just let x^5 = u in the taylor expansion of cos u, integrated term by term and you are left with an even nicer result than what mathematicia gives out.