Use integration by parts to evaluate $\int x ^ { 2 } e ^ { - 3 x } d x$ . Note that evaluation may require integration by parts more than once.
A) $- \frac { \left( 2 + 6 x + 9 x ^ { 2 } \right) e ^ { - 3 x } } { 27 } + C$
B) $\frac { \left( 1 + 6 x - 3 x ^ { 2 } \right) e ^ { - 3 x } } { 9 } + C$
C) $- \frac { \left( 2 + 9 x + 9 x ^ { 2 } \right) e ^ { - 3 x } } { 27 } + C$
D) $\frac { \left( 2 - 3 x + 9 x ^ { 2 } \right) e ^ { - 3 x } } { 9 } + C$
E) $- \frac { \left( 1 - 3 x + 9 x ^ { 2 } \right) e ^ { - 3 x } } { - 27 } + C$

Question

Quizplus · Accepted Answer

Integration by parts is applied multiple times, using $u = x^2$ and $dv = e^{-3x}dx$ initially, and then repeating the process with the resulting integrals. The correct result is $- \frac { \left( 2 + 6 x + 9 x ^ { 2 } \right) e ^ { - 3 x } } { 27 } + C$.

Use Integration by Parts to Evaluate ∫x2e−3xdx\int x ^ { 2 } e ^ { - 3 x } d x∫x2e−3xdx

Use Integration by Parts to Evaluate $\int x ^ { 2 } e ^ { - 3 x } d x$