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