Evaluate the Integral integral of xe^x with respect to x

Problem

(∫_^)(x*ex*d(x))

Solution

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

2. Choose the variables for substitution by letting u=x and d(v)=ex*d(x)

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

4. Substitute these values into the integration by parts formula:

(∫_^)(x*ex*d(x))=x*ex−(∫_^)(ex*d(x))

5. Evaluate the remaining integral (∫_^)(ex*d(x))=ex

6. Combine the terms and add the constant of integration C

(∫_^)(x*ex*d(x))=x*ex−ex+C

Final Answer

(∫_^)(x*ex*d(x))=x*ex−ex+C

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