How to evaluate $\int {\frac{{\cos {x^3}}}{x}dx}$

Question

How to evaluate $$\int {\frac{{\cos {x^3}}}{x}dx}?$$ Maple evaluates this as $$\frac{{{\text{Ci}}({x^3})}}{3}.$$ Edit: If this cannot be evaluated in terms of elementary functions, is there a general strategy which allows us to deduce that this is the case?

Concerning the edit : just some days earlier we had the same kind of question. Set $z=x^3$ to get $\ \frac 13\int \frac{\cos(z)}z dz$ and use the demonstration there for $\int \frac{\sin(z)}z dz$. — Raymond Manzoni, Nov 20 '12 at 09:09

score 3 · Accepted Answer · 2012-11-20T04:27:38.337

3

$$I = \int \dfrac{\cos(x^3)}{x} dx = \dfrac13 \int \dfrac{\cos(x^3)}{x^3} (3x^2)dx = \dfrac13 \int \dfrac{\cos(x^3)}{x^3} d(x^3) = \dfrac{\text{Ci}(x^3)}{3} + \text{constant}$$ There is no expression for the above integral in terms of "elementary functions". If the limits of the integral are from $-a$ to $a$, the Cauchy principal value of the integral is $0$ since the integrand is an odd function.

edited Nov 20 '12 at 04:27

answered Nov 20 '12 at 04:11

Yes, but what is that in terms of elementary functions? – glebovg Nov 20 '12 at 04:21
Can someone explain what is "Ci"? – learner Mar 29 '13 at 13:00
@learner $\operatorname{Ci}(\xi)$ stands for cosine integral. – glebovg Apr 05 '13 at 22:01
@glebovg thanks a lot. – learner Apr 06 '13 at 03:32

score 3 · Answer 2 · answered Nov 20 '12 at 08:36

3

http://reference.wolfram.com/mathematica/ref/CosIntegral.html

http://www.wolframalpha.com/input/?i=cosineintegral%28x%29

That means there is a real part and a imaginary part.

Disclaimer: I am in no way affiliated with Wolfram or Wolframalpha.

answered Nov 20 '12 at 08:36

r4d1um

131

How to evaluate $\int {\frac{{\cos {x^3}}}{x}dx}$

2 Answers2