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

Problem

(∫_^)(e^(x^2)*d(x))

Solution

1. Identify the type of integral. The function e^(x^2) does not have an elementary antiderivative in terms of standard algebraic, trigonometric, or logarithmic functions.

2. Relate the integral to the error function, erf(x) which is defined as 2/√(,π)*(∫_0^x)(e^(−t^2)*d(t))

3. Apply the power series expansion for e^u where u=x^2 to express the integral as an infinite series.

4. Substitute e^(x^2)=(∑_n=0^∞)(((x^2)^n)/(n!))=(∑_n=0^∞)((x^(2*n))/(n!))

5. Integrate the series term by term with respect to x

6. Express the result using the imaginary error function, erfi(x) which is defined such that its derivative is 2/√(,π)*e^(x^2)

Final Answer

(∫_^)(e^(x^2)*d(x))=√(,π)/2*erfi(x)+C

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