Evaluate the Integral integral of x^2e^(9x) with respect to x

Problem

(∫_^)(x2*e(9*x)*d(x))

Solution

1. Identify the method of integration by parts, which states (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x2 and d(v)=e(9*x)*d(x)

2. Differentiate u to find d(u)=2*x*d(x) and integrate d(v) to find v=1/9*e(9*x)

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

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

4. Identify that the new integral (∫_^)(2/9*x*e(9*x)*d(x)) requires integration by parts again. Let u=2/9*x and d(v)=e(9*x)*d(x)

5. Differentiate u to find d(u)=2/9*d(x) and integrate d(v) to find v=1/9*e(9*x)

6. Apply the integration by parts formula to the inner integral.

(∫_^)(2/9*x*e(9*x)*d(x))=2/81*x*e(9*x)−(∫_^)(2/81*e(9*x)*d(x))

7. Integrate the remaining term (∫_^)(2/81*e(9*x)*d(x))

(∫_^)(2/81*e(9*x)*d(x))=2/729*e(9*x)

8. Substitute the results back into the original expression and add the constant of integration C

(∫_^)(x2*e(9*x)*d(x))=1/9*x2*e(9*x)−2/81*x*e(9*x)+2/729*e(9*x)+C

Final Answer

(∫_^)(x2*e(9*x)*d(x))=1/9*x2*e(9*x)−2/81*x*e(9*x)+2/729*e(9*x)+C

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