Evaluate the Integral integral of xe^(-x) with respect to 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 an exponential function e(−x) use integration by parts.

2. Assign variables for the integration by parts 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) and integrate d(v) to find v=−e(−x)

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

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

5. Simplify the expression and the remaining integral.

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

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

(∫_^)(x*e(−x)*d(x))=−x*e(−x)−e(−x)+C

7. Factor out the common term −e(−x) to simplify the result.

(∫_^)(x*e(−x)*d(x))=−e(−x)*(x+1)+C

Final Answer

(∫_^)(x*e(−x)*d(x))=−e(−x)*(x+1)+C

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