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

Problem

(∫_^)(x2*e(3*x)*d(x))

Solution

1. Identify the method of integration by parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose u=x2 and d(v)=e(3*x)*d(x) which implies d(u)=2*x*d(x) and v=1/3*e(3*x)

3. Apply the integration by parts formula for the first time.

(∫_^)(x2*e(3*x)*d(x))=1/3*x2*e(3*x)−(∫_^)(2/3*x*e(3*x)*d(x))

4. Apply integration by parts again to the remaining integral (∫_^)(2/3*x*e(3*x)*d(x)) by choosing u=2/3*x and d(v)=e(3*x)*d(x)

5. Calculate the new components d(u)=2/3*d(x) and v=1/3*e(3*x)

6. Substitute these back into the expression.

(∫_^)(x2*e(3*x)*d(x))=1/3*x2*e(3*x)−(2/9*x*e(3*x)−(∫_^)(2/9*e(3*x)*d(x)))

7. Evaluate the final integral (∫_^)(2/9*e(3*x)*d(x))=2/27*e(3*x)

8. Simplify the expression and add the constant of integration C

(∫_^)(x2*e(3*x)*d(x))=1/3*x2*e(3*x)−2/9*x*e(3*x)+2/27*e(3*x)+C

Final Answer

(∫_^)(x2*e(3*x)*d(x))=(e(3*x))/27*(9*x2−6*x+2)+C

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