Evaluate the improper integral or state that it is divergent.
-$$\int _ { - \infty } ^ { 0 } 4 e ^ { x } \sin x d x$$
A) -4
B) 0
C) - 2
D) 2

Question

Quizplus · Accepted Answer

We can use integration by parts:
\begin{aligned}\int_{-\infty}^04 e^x \sin x dx & = \lim_{t	o-\infty}\int_t^04 e^x \sin x dx \ & = \lim_{t	o-\infty}\left[-4 e^x \cos x - 4 e^x \sin x ight]_{t}^{0} \ & = \lim_{t	o-\infty}\left(-4 + 4 e^{t} \cos t + 4 e^{t} \sin tight) \ & = -4 \end{aligned} 
Therefore, the best choice is C (-2), as none of the other answer choices are equal to -4.

Evaluate the Improper Integral or State That It Is Divergent ∫−∞04exsin⁡xdx\int _ { - \infty } ^ { 0 } 4 e ^ { x } \sin x d x∫−∞0​4exsinxdx

Evaluate the Improper Integral or State That It Is Divergent $\int _ { - \infty } ^ { 0 } 4 e ^ { x } \sin x d x$