積分公式-02

積分公式について，まとめてみました．

・オイラーの公式

・\( \Large \displaystyle y = \int e^{- \alpha t} \cdot sin ( \omega t) \ dt \)

\( \Large \displaystyle sin ( \omega t) = \frac{e^{j \omega t}-e^{-j \omega t}}{2j} \)

\( \Large \displaystyle y = \int e^{- \alpha t} \cdot \frac{e^{j \omega t}-e^{-j \omega t}}{2j} \ dt\)

\( \Large \displaystyle = \frac{1}{2j} \int \left[ e^{( - \alpha + j \omega)t} -e^{( - \alpha - j \omega)t}\right] \ dt \)

\( \Large \displaystyle = \frac{1}{2j} \left[ \frac{1}{- \alpha + j \omega} e^{( - \alpha + j \omega)t} - \frac{1}{- \alpha - j \omega} e^{( - \alpha - j \omega)t}\right] \)

\( \Large \displaystyle = \frac{1}{2j} \ e^{ - \alpha t} \ \left[ \frac{1}{- \alpha + j \omega} e^{j \omega t} - \frac{1}{- \alpha - j \omega} e^{ - j \omega t}\right] \)

\( \Large \displaystyle = \frac{1}{2j} \ e^{ - \alpha t} \ \left[ \frac{1}{- \alpha + j \omega} e^{j \omega t} + \frac{1}{ \alpha + j \omega} e^{ - j \omega t}\right] \)

\( \Large \displaystyle = \frac{1}{2j} \ e^{ - \alpha t} \ \left[ \frac{- \alpha - j \omega}{ \alpha^2 + \omega^2} e^{j \omega t} + \frac{ \alpha - j \omega}{ \alpha^2 + \omega^2} e^{ - j \omega t}\right] \)

\( \Large \displaystyle = \frac{1}{2j} \frac{ 1}{ \alpha^2 + \omega^2} \ e^{ - \alpha t} \ \left[ - \alpha ( e^{j \omega t} - e^{-j \omega t}) - j \ \omega( e^{j \omega t} + e^{-j \omega t}) \right] \)

\( \Large \displaystyle = \frac{ 1}{ \alpha^2 + \omega^2} \ e^{ - \alpha t} \ \left[ - \alpha \cdot sin ( \omega t) - \omega \cdot cos (\omega t) \right] \)

・\( \Large \displaystyle y = \int e^{- \alpha t} \cdot cos ( \omega t) \ dt \)

\( \Large \displaystyle cos ( \omega t) = \frac{e^{j \omega t}+e^{-j \omega t}}{2} \)

\( \Large \displaystyle y = \int e^{- \alpha t} \cdot \frac{e^{j \omega t}+e^{-j \omega t}}{2} \ dt\)

\( \Large \displaystyle = \frac{1}{2} \int \left[ e^{( - \alpha + j \omega)t} +e^{( - \alpha - j \omega)t}\right] \ dt \)

\( \Large \displaystyle = \frac{1}{2} \left[ \frac{1}{- \alpha + j \omega} e^{( - \alpha + j \omega)t} + \frac{1}{- \alpha - j \omega} e^{( - \alpha - j \omega)t}\right] \)

\( \Large \displaystyle = \frac{1}{2} \ e^{ - \alpha t} \ \left[ \frac{1}{- \alpha + j \omega} e^{j \omega t} + \frac{1}{- \alpha - j \omega} e^{ - j \omega t}\right] \)

\( \Large \displaystyle = \frac{1}{2} \ e^{ - \alpha t} \ \left[ \frac{1}{- \alpha + j \omega} e^{j \omega t} - \frac{1}{ \alpha + j \omega} e^{ - j \omega t}\right] \)

\( \Large \displaystyle = \frac{1}{2} \ e^{ - \alpha t} \ \left[ \frac{- \alpha - j \omega}{ \alpha^2 + \omega^2} e^{j \omega t} - \frac{ \alpha - j \omega}{ \alpha^2 + \omega^2} e^{ - j \omega t}\right] \)

\( \Large \displaystyle = \frac{1}{2} \frac{ 1}{ \alpha^2 + \omega^2} \ e^{ - \alpha t} \ \left[ - \alpha ( e^{j \omega t} + e^{-j \omega t}) - j \ \omega( e^{j \omega t} - e^{-j \omega t}) \right] \)

\( \Large \displaystyle = \frac{1}{2} \frac{ 1}{ \alpha^2 + \omega^2} \ e^{ - \alpha t} \ \left[ - \alpha ( e^{j \omega t} + e^{-j \omega t}) + \frac{1}{ j} \ \omega( e^{j \omega t} - e^{-j \omega t}) \right] \)

\( \Large \displaystyle = \frac{ 1}{ \alpha^2 + \omega^2} \ e^{ - \alpha t} \ \left[ - \alpha \cdot cos ( \omega t) + \omega \cdot sin (\omega t) \right] \)