~~REVEAL~~


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| $\;$ \\ $\;$ |$U_{\rm O} = f(U_{\rm I})$  |
| $\;$ \\ $\;$ | with III.|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ |$U_{\rm O}=\color{blue}{-U_{\rm D}}-U_C$  |
| $\;$ \\ $\;$ |with II.  and I.:$ \color{blue}{U_{\rm D}} = { 1 \over A_D } \cdot U_{\rm O} \overset{A_D -> \infty}\longrightarrow 0$|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ |$U_{\rm O}= \quad  0 \quad -\color{blue}{U_C}$|
| $\;$ \\ $\;$ |with V.: $\color{blue}{U_C}={ 1 \over C }\cdot(\int_{t_0}^{t_1} I_C \ {\rm d} t+ Q_0(t_0))$|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ |$U_{\rm O} = {-{ 1 \over C }\cdot}(\int_{t_0}^{t_1} \color{blue}{I_C} \ {\rm d} t+ Q_0(t_0)) $|
| $\;$ \\ $\;$ |with IV.: $\color{blue}{I_C}=I_R$|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ |$U_{\rm O} = \color{blue}{-{ 1 \over C }\cdot(}\int_{t_0}^{t_1} I_R \ {\rm d} t+ Q_0(t_0)\color{blue}{)} $|
| $\;$ \\ $\;$ |Factor out| 
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over C }\cdot\int_{t_0}^{t_1} I_R \ {\rm d} t - \color{blue}{ Q_0(t_0) \over C } $|
| $\;$ \\ $\;$ |integration constant: $\color{blue}{ Q_0(t_0) \over C }= U_C(t_0) = -U_{\rm O0}$|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over C }\cdot\int_{t_0}^{t_1} \color{blue}{I_R} \ {\rm d} t + U_{\rm O0}$|
| $\;$ \\ $\;$ |with VI. and II.: $\color{blue}{I_R}={ U_R \over R}={ U_{\rm I} \over R} $|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over C }\cdot\int_{t_0}^{t_1} \color{blue}{1 \over R} \cdot U_{\rm I} \ {\rm d} t + U_{\rm O0}$|
| $\;$ \\ $\;$ |move constant ahead| 
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} U_{\rm I} \ {\rm d} t + U_{\rm O0}$|
| $\;$ \\ $\;$ | insert time constant  $\tau = R \cdot C$ | 
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over {\tau} }\cdot\int_{t_0}^{t_1} U_{\rm I} \ {\rm d} t + U_{\rm O0}$| 
| $\;$ \\ $\;$ | |
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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