why e^x^2 is not intigrable ?.

Dear Sourabh,

There is no elementary function whose derivative is e−x2. By elementary function we mean something obtained using arithmetical operations and composition from the standard functions we all know and love. But this is not a serious problem. A few important definite integrals involving e−x2 have pleasant closed form.

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but how can we prove it mathamatically?

Actually we can integrate that function but involves a higher level of calculus.