Evaluate the Integral

Problem

(∫_^)(3*x2*e(x3)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the exponent x3 is 3*x2 which is present in the integrand.

2. Substitute u=x3

3. Differentiate u with respect to x to find d(u)=3*x2*d(x)

4. Rewrite the integral in terms of u by replacing 3*x2*d(x) with d(u) and x3 with u

(∫_^)(eu*d(u))

5. Integrate the exponential function with respect to u

eu+C

6. Back-substitute the original expression for u to return to the variable x

Final Answer

(∫_^)(3*x2*e(x3)*d(x))=e(x3)+C

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