Evaluate $\int ( 2 - 3 x ) e ^ { - 2 x } d x$ .
A) $- 3 e ^ { - 2 x } + C$
B) $\frac { 3 } { 2 } x e ^ { - 2 x } + C$
C) $e ^ { - 2 x } \left( \frac { 3 } { 2 } x - \frac { 1 } { 4 } \right) + C$
D) $x e ^ { - 2 x } + C$

Question

Quizplus · Accepted Answer

This integral can be solved using integration by parts, where $u = 2 - 3x$ and $dv = e^{-2x}dx$. The derivative of $u$ is $du = -3dx$, and the integral of $dv$ is $v = -\frac{1}{2}e^{-2x}$. Applying the integration by parts formula, $\int u dv = uv - \int v du$, gives $e^{-2x}\left(\frac{3}{2}x - \frac{1}{4}\right) + C$, which matches option C.

Evaluate ∫(2−3x)e−2xdx\int ( 2 - 3 x ) e ^ { - 2 x } d x∫(2−3x)e−2xdx

Evaluate $\int ( 2 - 3 x ) e ^ { - 2 x } d x$