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My books states that the integrals like $\int \frac{\sin x}{x}dx$ and $\int e^{x^2}dx$ exist but they cannot be easily evaluated by elementary functions.I feel it is more because I am unable to evaluate it but can someone please tell me if there is a closed form for them?

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en.wikipedia.org/wiki/Trigonometric_integral and en.wikipedia.org/wiki/Error_function should get you started. –  Kris Jul 23 '12 at 4:27
    
See this, this, and this, for starters. –  J. M. Jul 23 '12 at 4:54

1 Answer 1

Those two functions have no closed form antiderivative with only elementary functions. This is provable. It's sometimes possible to cleanly compute definite integrals involving such functions. For example,

$$\int_0^\infty \frac{\sin x}{x}\ d x = \frac{\pi}{2}.$$

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