Chapter 8: Techniques of Integration - Additional and Advanced Exercises - Page 521: 36

$$\frac{1}{9}\left( {4\cos 5x\cos 4x + 5\sin 4x\sin 5x} \right) + C$$

$\begin{gathered} \int {\cos 5x\sin 4x} dx \hfill \\ {\text{Use tabular integration to evaluate the integral}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}} {\cos 5x{\text{ and its derivatives}}}&{}&{\sin 4x{\text{ and its integrals}}} \\ {\cos 5x}&{\mathop {\left( + \right)}\limits_ \searrow }&{\sin 4x} \\ { - 5\sin 5x}&{\mathop {\left( - \right)}\limits_ \searrow }&{ - \frac{1}{4}\cos 4x} \\ { - 25\cos 5x}&{\left( + \right)}&{ - \frac{1}{{16}}\sin 4x} \end{array} \hfill \\ \end{gathered} $ $$\eqalign{ & {\text{We need stop here in the term }} - \frac{1}{{16}}\sin 4x{\text{ because it is the same as the first row except }} \cr & {\text{for constants }}\left( { - 25{\text{ on the left and }} - \frac{1}{{16}}{\text{ on the right}}} \right).{\text{ Then}}{\text{:}} \cr & {\text{We interpret the table as saying }} \cr & \int {\cos 5x\sin 4x} dx = \left( {\cos 5x} \right)\left( { - \frac{1}{4}\cos 4x} \right) - \left( { - 5\sin 5x} \right)\left( { - \frac{1}{{16}}\sin 4x} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \int {\left( { - 25\cos 5x} \right)\left( { - \frac{1}{{16}}\sin 4x} \right)} dx \cr & \int {\cos 5x\sin 4x} dx = - \frac{1}{4}\cos 5x\cos 4x - \frac{5}{{16}}\sin 4x\sin 5x + \frac{{25}}{{16}}\int {\cos 5x\sin 4x} dx \cr & \cr & {\text{Solve for }}\int {\cos 5x\sin 4x} dx \cr & \int {\cos 5x\sin 4x} dx - \frac{{25}}{{16}}\int {\cos 5x\sin 4x} dx = - \frac{1}{4}\cos 5x\cos 4x - \frac{5}{{16}}\sin 4x\sin 5x + C \cr & - \frac{9}{{16}}\int {\cos 5x\sin 4x} dx = - \frac{1}{4}\cos 5x\cos 4x - \frac{5}{{16}}\sin 4x\sin 5x + C \cr & \int {\cos 5x\sin 4x} dx = - \frac{{16}}{9}\left( { - \frac{1}{4}\cos 5x\cos 4x - \frac{5}{{16}}\sin 4x\sin 5x} \right) + C \cr & \int {\cos 5x\sin 4x} dx = \frac{1}{9}\left( {4\cos 5x\cos 4x +5\sin 4x\sin 5x} \right) + C \cr} $$