Find $\int e^{-t / 4} \sin 4 t d t$ .
A) $-\frac{16}{257} e^{-t / 4}\left(\frac{1}{4} \sin 4 t+4 \cos 4 t\right)+C$
B) $\frac{16}{257} e^{-t / 4}\left(\frac{1}{4} \sin 4 t-4 \cos 4 t\right)+C$
C) $-\frac{1}{32} e^{-t / 4}\left(\frac{1}{4} \sin 4 t+4 \cos 4 t\right)+C$
D) $\frac{1}{32} e^{-t / 4}\left(\frac{1}{4} \sin 4 t-4 \cos 4 t\right)+C$

Question

Quizplus · Accepted Answer

This integral can be solved using integration by parts or the tabular method, which involves differentiating $e^{-t/4}$ and integrating $\sin(4t)$ repeatedly. The correct solution involves a combination of $\sin(4t)$ and $\cos(4t)$ multiplied by $e^{-t/4}$, with coefficients that balance the derivatives and integrals correctly. The coefficients and signs in option A correctly reflect the outcome of such a process, taking into account the derivatives of $e^{-t/4}$ and the integrals of $\sin(4t)$, leading to the factor of $16/257$ and the specific combination of sine and cosine functions.

Find ∫e−t/4sin⁡4tdt\int e^{-t / 4} \sin 4 t d t∫e−t/4sin4tdt A)

Find $\int e^{-t / 4} \sin 4 t d t$
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