| $\;$ $\;$ $\;$ | $U_{\rm O} = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} \color{blue}{U_{\rm I}(t)} \ {\rm d}t + U_{\rm O0}$ |
| $\;$ $\;$ $\;$ | insert sine function: $ \color{blue}{U_{\rm I}(t)}= \hat{U}_{\rm I} \cdot \sin(\omega \cdot t)$ |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_{\rm O} = -{ 1 \over {R\cdot C} }\cdot\color{blue}{\int_{t_0}^{t_1} \hat{U}_{\rm I} \cdot \sin(\omega \cdot t) \ {\rm d}t} + U_{\rm O0}$ |
| $\;$ $\;$ $\;$ | insert root function with limits $\color{blue}{\int_{x_0}^{x_1} \sin(a \cdot x) \ {\rm d}x} = [- {1 \over a} \cdot \cos(a \cdot x) ]_{x_0}^{x_1}$ |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_{\rm O} = -{ 1 \over {R\cdot C} }\cdot [- \color{blue}{\hat{U}_{\rm I} \over \omega} \cdot \cos(\omega \cdot t) ]_{t_0}^{t_1} + U_{\rm O0}$ |
| $\;$ $\;$ $\;$ | put constant before integral |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_{\rm O} = { 1 \over {R\cdot C} }\cdot {\hat{U}_{\rm I} \over \omega} \cdot \color{blue}{[ \cos(\omega \cdot t) ]_{t_0}^{t_1}} + U_{\rm O0}$ |
| $\;$ $\;$ $\;$ | insert limits: $t_0=0$, $t_1=t$ |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_{\rm O} = {{{\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot (} \cos(\omega \cdot t) - \color{blue}{\cos(0)} ) + U_{\rm O0}$ |
| $\;$ $\;$ $\;$ | $\color{blue}{\cos(0)}=1$ |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_{\rm O} = \color{blue}{{{ \hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot (} \cos(\omega \cdot t) - 1 \color{blue}{)} + U_{\rm O0}$ |
| $\;$ $\;$ $\;$ | multiply |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_{\rm O} = { {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot \cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C}} + U_{\rm O0}}$ |
| $\;$ $\;$ $\;$ | consider the non-cosine terms: The blue part is independent in time. We assume purely sinusoidal quantities! |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_{\rm O} = { {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot \cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C}} + U_{\rm O0}}$ | |
| $\;$ $\;$ $\;$ | $\rightarrow$ initial voltage of the capacitor: $U_{C0} = U_{\rm O0}={{\hat{U}_{\rm I}} \over {\omega \cdot R\cdot C}}$ |
|
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ | ||
| $\;$ $\;$ $\;$ | $U_{\rm O} = { {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot \cos(\omega \cdot t)$ |
| $\;$ $\;$ $\;$ | |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |