Evaluate the Integral integral of x^2e^(8x) with respect to x

Problem

(∫_^)(x2*e(8*x)*d(x))

Solution

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

2. Choose u=x2 and d(v)=e(8*x)*d(x)

3. Differentiate u to find d(u)=2*x*d(x) and integrate d(v) to find v=1/8*e(8*x)

4. Apply the integration by parts formula for the first time:

(∫_^)(x2*e(8*x)*d(x))=1/8*x2*e(8*x)−(∫_^)(1/4*x*e(8*x)*d(x))

5. Apply integration by parts again to the remaining integral (∫_^)(1/4*x*e(8*x)*d(x)) by choosing u=1/4*x and d(v)=e(8*x)*d(x)

6. Differentiate u to find d(u)=1/4*d(x) and integrate d(v) to find v=1/8*e(8*x)

7. Substitute these values back into the formula:

(∫_^)(1/4*x*e(8*x)*d(x))=1/32*x*e(8*x)−(∫_^)(1/32*e(8*x)*d(x))

8. Evaluate the final integral:

(∫_^)(1/32*e(8*x)*d(x))=1/256*e(8*x)

9. Combine all terms and add the constant of integration C

(∫_^)(x2*e(8*x)*d(x))=1/8*x2*e(8*x)−1/32*x*e(8*x)+1/256*e(8*x)+C

Final Answer

(∫_^)(x2*e(8*x)*d(x))=(e(8*x))/8*(x2−x/4+1/32)+C

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