Evaluate the Integral

Problem

(∫_^)(e(x2)/2*d(x))

Solution

1. Identify the integral as a non-elementary integral. The function ƒ(x)=e(x2)/2 does not have an antiderivative that can be expressed in terms of basic functions like polynomials, logarithms, or trigonometric functions.

2. Relate the integral to the imaginary error function, erfi(z) which is defined as erfi(z)=2/√(,π)*(∫_0^z)(e(t2)*d(t))

3. Perform a substitution to match the standard form. Let u=x/√(,2) which implies x=u√(,2) and d(x)=√(,2)*d(u)

4. Substitute these into the integral to get (∫_^)(e(u2)√(,2)*d(u))

5. Apply the definition of the imaginary error function, where (∫_^)(e(u2)*d(u))=√(,π)/2*erfi(u)+C

6. Back-substitute u=x/√(,2) to find the final expression.

Final Answer

(∫_^)(e(x2)/2*d(x))=√(,π/2)*erfi*(x/√(,2))+C

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