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

Problem

(∫_^)(x2*e(−3*x)*d(x))

Solution

1. Identify the integration method as 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)

3. Calculate the differentials d(u)=2*x*d(x) and v=(∫_^)(e(−3*x)*d(x))=−1/3*e(−3*x)

4. 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))

5. Simplify the expression before the second integration.

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

6. Apply integration by parts again for the remaining integral (∫_^)(x*e(−3*x)*d(x)) with u=x and d(v)=e(−3*x)*d(x)

7. Calculate the new differentials d(u)=d(x) and v=−1/3*e(−3*x)

8. Substitute the result of the second integration by parts back into the equation.

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

9. Evaluate the final integral (∫_^)(e(−3*x)*d(x))=−1/3*e(−3*x)

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

10. Factor out the common term −1/27*e(−3*x) to simplify the final expression.

Final Answer

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

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