Find the Antiderivative e^(-x^2)

Problem

(∫_^)(e(−x2)*d(x))

Solution

1. Identify the integral as the Gaussian integral form, which does not have an antiderivative expressible in terms of elementary functions (such as polynomials, logarithms, or trigonometric functions).

2. Define the result using the error function, denoted as erf(x) which is specifically defined to represent the area under the bell curve.

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

4. Adjust the constants to match the integrand e(−x2) resulting in a factor of √(,π)/2 multiplying the error function.

5. Add the constant of integration C to complete the general antiderivative.

Final Answer

(∫_^)(e(−x2)*d(x))=√(,π)/2*erf(x)+C

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