$3$\displaystyle{ I=\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{e}^{ ax }\sin bx- \frac{\hspace{2}b\hspace{2}}{\hspace{2}a\hspace{2}}J }$ より
$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{e}^{ ax }\sin bx}$
　 $3$\displaystyle{- \frac{\hspace{2}b\hspace{2}}{\hspace{2}a\hspace{2}} $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{e}^{ ax }\cos bx+\frac{\hspace{2}b\hspace{2}}{\hspace{2}a\hspace{2}}I $ }$
$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{e}^{ ax }\sin bx}$
　 $3$\displaystyle{- \frac{\hspace{2}b\hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{e}^{ ax }\cos bx- \frac{\hspace{2} {b}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}I}$
$3$\displaystyle{\frac{\hspace{2} {a}^{2}+{b}^{2} \hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}I}$
$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{e}^{ ax }\sin bx- \frac{\hspace{2}b\hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{e}^{ ax }\cos bx}$
$3$\displaystyle{I=\frac{\hspace{2} {a}^{2} \hspace{2}}{\hspace{2} {a}^{2}+{b}^{2} \hspace{2}}}$
　 $3$\displaystyle{ $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}a\hspace{2}}{e}^{ ax }\sin bx- \frac{\hspace{2}b\hspace{2}}{\hspace{2} {a}^{2} \hspace{2}}{e}^{ ax }\cos bx $ }$
　 $3$\displaystyle{=\frac{\hspace{2} {e}^{ ax } \hspace{2}}{\hspace{2} {a}^{2}+{b}^{2} \hspace{2}} $ a\sin bx- b\cos bx $ +C}$
　　　　　　　　　　　　　　　　( $3$\displaystyle{C}$ は積分定数）

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