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

Problem

(∫_^)(x2*ex*d(x))

Solution

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

2. Assign variables for the first application of integration by parts by letting u=x2 and d(v)=ex*d(x)

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

4. Substitute these 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)) by letting u=2*x and d(v)=ex*d(x)

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

7. Substitute these into the second integration step.

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

8. Evaluate the final integral (∫_^)(2*ex*d(x))=2*ex

9. Combine all parts and simplify the expression, adding the constant of integration C

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

(∫_^)(x2*ex*d(x))=x2*ex−2*x*ex+2*ex+C

Final Answer

(∫_^)(x2*ex*d(x))=ex*(x2−2*x+2)+C

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