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