Chapter 8: Techniques of Integration - Additional and Advanced Exercises - Page 521: 34

$$\frac{{{e^{3x}}}}{{25}}\left( {3\sin 4x - 4\cos 4x} \right) + C$$

$\begin{gathered} \int {{e^{3x}}\sin 4x} dx \hfill \\ \hfill \\ {\text{Use tabular integration to evaluate the integral}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}} {{e^{3x}}{\text{ and its derivatives}}}&{}&{\sin 4x{\text{ and its integrals}}} \\ {{e^{3x}}}&{\mathop {\left( + \right)}\limits_ \searrow }&{\sin 4x} \\ {3{e^{3x}}}&{\mathop {\left( - \right)}\limits_ \searrow }&{ - \frac{1}{4}\cos 4x} \\ {9{e^{3x}}}&{\left( + \right)}&{ - \frac{1}{{16}}\sin 4x} \end{array} \hfill \\ \end{gathered} $ $$\eqalign{ & {\text{We need to stop here because it is the same as the first row except }} \cr & {\text{for constants }}\left( {{\text{9 on the left and }} - \frac{1}{{16}}{\text{ on the right}}} \right).{\text{ Then}}{\text{:}} \cr & {\text{We interpret the table as saying }} \cr & \int {{e^{3x}}\sin 4x} dx = \left( {{e^{3x}}} \right)\left( { - \frac{1}{4}\cos 4x} \right) - \left( {3{e^{3x}}} \right)\left( { - \frac{1}{{16}}\sin 4x} \right) + \int {\left( {9{e^{3x}}} \right)\left( { - \frac{1}{{16}}\sin 4x} \right)} dx \cr & \int {{e^{3x}}\sin 4x} dx = - \frac{1}{4}{e^{3x}}\cos 4x + \frac{3}{{16}}{e^{3x}}\sin 4x + \frac{9}{{16}}\int {{e^{3x}}\sin 4x} dx \cr & \cr & {\text{Solve for }}\int {{e^{3x}}\sin 4x} dx \cr & \int {{e^{3x}}\sin 4x} dx + \frac{9}{{16}}\int {{e^{3x}}\sin 4x} dx = - \frac{1}{4}{e^{3x}}\cos 4x + \frac{3}{{16}}{e^{3x}}\sin 4x + C \cr & \frac{{25}}{{16}}\int {{e^{3x}}\sin 4x} dx = - \frac{1}{4}{e^{3x}}\cos 4x + \frac{3}{{16}}{e^{3x}}\sin 4x + C \cr & \int {{e^{3x}}\sin 4x} dx = \frac{{16}}{{25}}\left( { - \frac{1}{4}{e^{3x}}\cos 4x + \frac{3}{{16}}{e^{3x}}\sin 4x} \right) + C \cr & \int {{e^{3x}}\sin 4x} dx = \frac{{{e^{3x}}}}{{25}}\left( {3\sin 4x - 4\cos 4x} \right) + C \cr} $$