Evaluate the Integral integral of x^2e^(7x) with respect to x

Problem

(∫_^)(x2*e(7*x)*d(x))

Solution

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

2. Choose u=x2 and d(v)=e(7*x)*d(x)

3. Calculate the differentials d(u)=2*x*d(x) and v=1/7*e(7*x)

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

(∫_^)(x2*e(7*x)*d(x))=1/7*x2*e(7*x)−(∫_^)(2/7*x*e(7*x)*d(x))

5. Apply integration by parts again to the remaining integral (∫_^)(2/7*x*e(7*x)*d(x)) by choosing u=2/7*x and d(v)=e(7*x)*d(x)

6. Calculate the new differentials d(u)=2/7*d(x) and v=1/7*e(7*x)

7. Substitute these values back into the expression.

(∫_^)(x2*e(7*x)*d(x))=1/7*x2*e(7*x)−(2/49*x*e(7*x)−(∫_^)(2/49*e(7*x)*d(x)))

8. Evaluate the final integral (∫_^)(2/49*e(7*x)*d(x))=2/343*e(7*x)

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

(∫_^)(x2*e(7*x)*d(x))=1/7*x2*e(7*x)−2/49*x*e(7*x)+2/343*e(7*x)+C

Final Answer

(∫_^)(x2*e(7*x)*d(x))=(e(7*x))/7*(x2−(2*x)/7+2/49)+C

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