Evaluate the integral.
-$\int y ^ { 3 } e ^ { - 3 y } d y$
A) $- \mathrm { e } ^ { - 3 \mathrm { y } } \left[ \frac { 1 } { 3 } \mathrm { y } ^ { 3 } + \frac { 1 } { 3 } \mathrm { y } ^ { 2 } + \frac { 2 } { 9 } \mathrm { y } + \frac { 2 } { 27 } \right] + \mathrm { C }$
B) $\mathrm { e } ^ { - 3 \mathrm { y } } \left[ \frac { 1 } { 3 } \mathrm { y } ^ { 3 } - \frac { 1 } { 3 } \mathrm { y } ^ { 2 } + \frac { 2 } { 9 } \mathrm { y } - \frac { 2 } { 27 } \right] + \mathrm { C }$
C) $- \frac { 1 } { 12 } y ^ { 4 } e ^ { - 3 y } + C$
D) $- \frac { 1 } { 3 } e ^ { - 3 y } \left[ y ^ { 3 } + y ^ { 2 } + y + 6 \right] + C$

Question

Quizplus · Accepted Answer

This integral can be solved using integration by parts or the tabular method, repeatedly applying the technique to reduce the power of $y$ each time. The correct result involves a combination of $y^3$, $y^2$, $y$, and constant terms, all multiplied by $e^{-3y}$, with coefficients that ensure each differentiation step according to the power of $y$ is accounted for correctly. Option A correctly represents this process and its result, including the negative sign outside which comes from integrating $e^{-3y}$.

Evaluate the Integral ∫y3e−3ydy\int y ^ { 3 } e ^ { - 3 y } d y∫y3e−3ydy

Evaluate the Integral $\int y ^ { 3 } e ^ { - 3 y } d y$