Evaluate the Integral integral of z/(e^z) with respect to z

Problem

(∫_^)(z/(ez)*d(z))

Solution

1. Rewrite the integrand using a negative exponent to prepare for integration by parts.

(∫_^)(z*e(−z)*d(z))

2. Identify the parts for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

u=z

d(v)=e(−z)*d(z)

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

d(u)=d(z)

v=−e(−z)

4. Apply the formula for integration by parts.

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

5. Simplify the expression and the remaining integral.

−z*e(−z)+(∫_^)(e(−z)*d(z))

6. Evaluate the final integral and add the constant of integration C

−z*e(−z)−e(−z)+C

7. Factor out the common term to simplify the final expression.

−e(−z)*(z+1)+C

Final Answer

(∫_^)(z/(ez)*d(z))=−(z+1)/(ez)+C

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