Evaluate the Integral integral of x^2e^(x^3) with respect to x

Problem

(∫_^)(x2*e(x3)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the exponent x3 is a multiple of the x2 term present in the integrand.

2. Substitute u=x3 to simplify the expression.

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=3*x2

4. Solve for the differential x2*d(x) to prepare for substitution into the integral.

x2*d(x)=1/3*d(u)

5. Rewrite the integral in terms of u by substituting the expressions found in the previous steps.

(∫_^)(eu⋅1/3*d(u))

6. Factor out the constant 1/3 from the integral.

1/3*(∫_^)(eu*d(u))

7. Integrate the exponential function eu with respect to u

1/3*eu+C

8. Back-substitute the original expression x3 for u to obtain the final result in terms of x

1/3*e(x3)+C

Final Answer

(∫_^)(x2*e(x3)*d(x))=1/3*e(x3)+C

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