Evaluate the Integral integral of x^2e^(5x) with respect to x

Problem

(∫_^)(x2*e(5*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(5*x)*d(x) which implies d(u)=2*x*d(x) and v=1/5*e(5*x)

3. Apply the formula for the first time:

(∫_^)(x2*e(5*x)*d(x))=1/5*x2*e(5*x)−(∫_^)(2/5*x*e(5*x)*d(x))

4. Apply Integration by Parts again for the remaining integral (∫_^)(2/5*x*e(5*x)*d(x)) by choosing u=2/5*x and d(v)=e(5*x)*d(x)

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

6. Substitute these into the second integration step:

(∫_^)(2/5*x*e(5*x)*d(x))=2/25*x*e(5*x)−(∫_^)(2/25*e(5*x)*d(x))

7. Evaluate the final integral:

(∫_^)(2/25*e(5*x)*d(x))=2/125*e(5*x)

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

(∫_^)(x2*e(5*x)*d(x))=1/5*x2*e(5*x)−2/25*x*e(5*x)+2/125*e(5*x)+C

Final Answer

(∫_^)(x2*e(5*x)*d(x))=1/5*x2*e(5*x)−2/25*x*e(5*x)+2/125*e(5*x)+C

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