Find the Integral xe^(-x)

Problem

(∫_^)(x*e(−x)*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 the variables for the integration by parts formula by letting 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. Substitute these values into the integration by parts formula.

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

6. Simplify the expression and the integral.

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

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

(∫_^)(x*e(−x)*d(x))=−x*e(−x)−e(−x)+C

8. Factor out the common term −e(−x) to simplify the final 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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