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

Problem

(∫_^)(x2*ex*d(x))

Solution

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

2. Assign variables for the first application: let u=x2 and d(v)=ex*d(x)

3. Differentiate and integrate to find d(u)=2*x*d(x) and v=ex

4. Substitute into the integration by parts formula:

(∫_^)(x2*ex*d(x))=x2*ex−(∫_^)(2*x*ex*d(x))

5. Apply integration by parts a second time to the remaining integral (∫_^)(2*x*ex*d(x))

6. Assign variables for the second application: let u=2*x and d(v)=ex*d(x)

7. Differentiate and integrate to find d(u)=2*d(x) and v=ex

8. Substitute these into the second integral:

(∫_^)(2*x*ex*d(x))=2*x*ex−(∫_^)(2*ex*d(x))

9. Evaluate the final simple integral:

(∫_^)(2*ex*d(x))=2*ex

10. Combine all parts and include the constant of integration C

(∫_^)(x2*ex*d(x))=x2*ex−(2*x*ex−2*ex)+C

11. Simplify the expression by distributing the negative sign and factoring out ex

(∫_^)(x2*ex*d(x))=ex*(x2−2*x+2)+C

Final Answer

(∫_^)(x2*ex*d(x))=ex*(x2−2*x+2)+C

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