Evaluate the Integral integral of z^3e^z with respect to z

Problem

(∫_^)(z^3*e^z*d(z))

Solution

1. Identify the method of integration by parts, which states (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Apply integration by parts repeatedly or use the tabular method (DI method) for a polynomial multiplied by an exponential.

3. Differentiate the polynomial part z^3 until it reaches zero: z^3→3*z^2→6*z→6→0

4. Integrate the exponential part e^z repeatedly: e^z→e^z→e^z→e^z→e^z

5. Combine the products of the diagonals with alternating signs: (+)*(z^3)*(e^z)+(−)*(3*z^2)*(e^z)+(+)*(6*z)*(e^z)+(−)*(6)*(e^z)

6. Factor out the common term e^z and add the constant of integration C

Final Answer

(∫_^)(z^3*e^z*d(z))=e^z*(z^3−3*z^2+6*z−6)+C

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