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

Problem

(∫_^)(e(x2)*d(x))

Solution

1. Identify the type of integral. The function e(x2) 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(−t2)*d(t))

3. Apply the power series expansion for eu where u=x2 to express the integral as an infinite series.

4. Substitute e(x2)=(∑_n=0^∞)(((x2)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(x2)

Final Answer

(∫_^)(e(x2)*d(x))=√(,π)/2*erfi(x)+C

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