Evaluate the Integral integral of e^(-x^2) with respect to x

Problem

(∫_^)(e(−x2)*d(x))

Solution

1. Identify the integral as the Gaussian integral, which does not have an antiderivative expressible in terms of elementary functions.

2. Define the solution using the error function, denoted as erf(x) which is specifically defined to represent this integral.

3. Apply the standard definition of the error function, where erf(x)=2/√(,π)*(∫_0^x)(e(−t2)*d(t))

4. Relate the indefinite integral to the error function by adjusting for the constant factor √(,π)/2 and adding the constant of integration C

Final Answer

(∫_^)(e(−x2)*d(x))=√(,π)/2*erf(x)+C

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