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

Problem

(∫_^)(x2*e(2*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(2*x)*d(x)

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

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

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

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

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

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

7. Evaluate the final simple integral.

(∫_^)(1/2*e(2*x)*d(x))=1/4*e(2*x)

8. Substitute the results back into the main 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. Simplify the expression by distributing the negative sign and factoring.

(∫_^)(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

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

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