Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Why downvote? Care to comment? – glebovg Nov 20 '12 at 4:20
You may need to read this. – Mhenni Benghorbal Nov 20 '12 at 5:00
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 9:09
up vote 3 down vote accepted

$$I = \int \dfrac{\cos(x^3)}{x} dx = \dfrac13 \int \dfrac{\cos(x^3)}{x^3} (3x^2)dx = \dfrac13 \int \dfrac{\cos(x^3)}{x^3} d(x^3) = \dfrac{\text{Ci}(x^3)}{3} + \text{constant}$$ 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.

share|cite|improve this answer
Yes, but what is that in terms of elementary functions? – glebovg Nov 20 '12 at 4:21
Can someone explain what is "Ci"? – learner Mar 29 '13 at 13:00
@learner $\operatorname{Ci}(\xi)$ stands for cosine integral. – glebovg Apr 5 '13 at 22:01
@glebovg thanks a lot. – learner Apr 6 '13 at 3:32

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

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.