Evaluate the Integral integral of ye^y with respect to y

Problem

(∫_^)(y*e^y*d(y))

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=y and d(v)=e^y*d(y)

3. Differentiate u to find d(u)=d(y) and integrate d(v) to find v=e^y

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

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

5. Evaluate the remaining integral (∫_^)(e^y*d(y))=e^y

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

y*e^y−e^y+C

7. Factor out the common term e^y to simplify the expression.

Final Answer

(∫_^)(y*e^y*d(y))=e^y*(y−1)+C

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