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

Problem

(∫_^)(e(2*x)*x2*d(x))

Solution

1. Identify the method of integration by parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x2 and d(v)=e(2*x)*d(x)

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

3. Apply the formula for the first time:

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

4. Apply integration by parts again for the remaining integral (∫_^)(x*e(2*x)*d(x)) Let u=x and d(v)=e(2*x)*d(x)

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

6. Substitute these into the second integration step:

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

7. Evaluate the final integral:

(∫_^)(1/2*e(2*x)*d(x))=1/4*e(2*x)

8. Combine all parts and simplify the expression:

(∫_^)(x2*e(2*x)*d(x))=1/2*x2*e(2*x)−(1/2*x*e(2*x)−1/4*e(2*x))+C

9. Distribute the negative sign and factor out common terms:

(∫_^)(x2*e(2*x)*d(x))=1/2*x2*e(2*x)−1/2*x*e(2*x)+1/4*e(2*x)+C

Final Answer

(∫_^)(e(2*x)*x2*d(x))=(e(2*x)*(2*x2−2*x+1))/4+C

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