Evaluate the Integral integral of e^(x^3) with respect to x

Problem

(∫_^)(e(x3)*d(x))

Solution

1. Identify the type of integral. The function e(x3) is a composition of an exponential function and a power function.

2. Determine if an elementary antiderivative exists. The integral (∫_^)(e(x3)*d(x)) is a well-known non-elementary integral, meaning its antiderivative cannot be expressed in terms of basic functions like polynomials, logarithms, or trigonometric functions.

3. Apply a power series expansion to find a representation of the integral. Recall the Taylor series for eu

eu=(∑_n=0^∞)((un)/(n!))

4. Substitute u=x3 into the series:

e(x3)=(∑_n=0^∞)(((x3)n)/(n!))

e(x3)=(∑_n=0^∞)((x(3*n))/(n!))

5. Integrate the series term by term:

(∫_^)((∑_n=0^∞)((x(3*n))/(n!))*d(x))=(∑_n=0^∞)((∫_^)((x(3*n))/(n!)*d(x)))

6. Evaluate the power rule for each term:

(∫_^)((x(3*n))/(n!)*d(x))=(x(3*n+1))/(n!(3*n+1))

7. Add the constant of integration C

Final Answer

(∫_^)(e(x3)*d(x))=(∑_n=0^∞)((x(3*n+1))/(n!(3*n+1)))+C

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