~~REVEAL~~

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| $\;$ \\ $\;$ \\ $\;$ |$U_A = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} \color{blue}{U_E(t)} \ dt + U_{A0}$|
| $\;$ \\ $\;$ \\ $\;$ | insert sine function: \\ $ \color{blue}{U_E(t)}= \hat{U}_E \cdot sin(\omega \cdot t)$|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ \\ $\;$ |$U_A = -{ 1 \over {R\cdot C} }\cdot\color{blue}{\int_{t_0}^{t_1} \hat{U}_E \cdot sin(\omega \cdot t) \ dt} + U_{A0}$|
| $\;$ \\ $\;$ \\ $\;$ |insert root function with limits \\  $\color{blue}{\int_{x_0}^{x_1} sin(a \cdot x) \ dx} = [- {1 \over a} \cdot cos(a \cdot x) ]_{x_0}^{x_1}$|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ \\ $\;$ |$U_A = -{ 1 \over {R\cdot C} }\cdot [- \color{blue}{\hat{U}_E \over \omega} \cdot cos(\omega \cdot t) ]_{t_0}^{t_1} + U_{A0}$ |
| $\;$ \\ $\;$ \\ $\;$ | put constant before integral|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ \\ $\;$ |$U_A = { 1 \over {R\cdot C} }\cdot {\hat{U}_E \over \omega} \cdot \color{blue}{[ cos(\omega \cdot t) ]_{t_0}^{t_1}} + U_{A0}$|
| $\;$ \\ $\;$ \\ $\;$ |insert limits: $t_0=0$, $t_1=t$|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ \\ $\;$ |$U_A = {{{\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - \color{blue}{cos(0)} ) + U_{A0}$|
| $\;$ \\ $\;$ \\ $\;$ | $\color{blue}{cos(0)}=1$|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ \\ $\;$ |$U_A = \color{blue}{{{ \hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - 1 \color{blue}{)} + U_{A0}$ |
| $\;$ \\ $\;$ \\ $\;$ |multiply|
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ \\ $\;$ |$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_E } \over {\omega \cdot R\cdot C}} + U_{A0}}$ |
| $\;$ \\ $\;$ \\ $\;$ |consider the non-cosine terms: \\ The blue part is independent in time. \\ We assume purely sinusoidal quantities! |
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ \\ $\;$ |$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_E } \over {\omega \cdot R\cdot C}} + U_{A0}}$ |
| $\;$ \\ $\;$ \\ $\;$ | $\rightarrow$ initial voltage of the capacitor: \\ $U_{C0} = U_{A0}={{\hat{U}_E} \over {\omega \cdot R\cdot C}}$||
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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| $\;$ \\ $\;$ \\ $\;$ |$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t)$| 
| $\;$ \\ $\;$ \\ $\;$ | |
| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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