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

Problem

(∫_^)(z3*ez*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 z3 until it reaches zero: z3→3*z2→6*z→6→0

4. Integrate the exponential part ez repeatedly: ez→ez→ez→ez→ez

5. Combine the products of the diagonals with alternating signs: (+)*(z3)*(ez)+(−)*(3*z2)*(ez)+(+)*(6*z)*(ez)+(−)*(6)*(ez)

6. Factor out the common term ez and add the constant of integration C

Final Answer

(∫_^)(z3*ez*d(z))=ez*(z3−3*z2+6*z−6)+C

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