Find the Integral xe^x

Problem

(∫_^)(x*e^x*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function x and a transcendental function e^x use integration by parts.

2. Choose the parts for the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x and d(v)=e^x*d(x)

3. Differentiate u to find d(u)=d(x)

4. Integrate d(v) to find v=e^x

5. Apply the integration by parts formula.

(∫_^)(x*e^x*d(x))=x*e^x−(∫_^)(e^x*d(x))

6. Evaluate the remaining integral (∫_^)(e^x*d(x))=e^x

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

x*e^x−e^x+C

8. Factor out the common term e^x to simplify the expression.

(x−1)*e^x+C

Final Answer

(∫_^)(x*e^x*d(x))=(x−1)*e^x+C

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