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

Problem

(∫_^)(x*e(3*x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the exponent 3*x2 has a derivative that is a multiple of the x term outside the exponential.

2. Substitute a new variable u for the exponent.

u=3*x2

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

d(u)/d(x)=6*x

d(u)=6*x*d(x)

4. Solve for the terms present in the original integral, which are x*d(x)

1/6*d(u)=x*d(x)

5. Rewrite the integral in terms of u by substituting 3*x2 with u and x*d(x) with 1/6*d(u)

(∫_^)(1/6*eu*d(u))

6. Integrate the expression with respect to u

1/6*eu+C

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

1/6*e(3*x2)+C

Final Answer

(∫_^)(x*e(3*x2)*d(x))=1/6*e(3*x2)+C

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