I need to find Vout in time domain for below circuit.
Here's my attempt, using Laplace transform.
$$V_{in}(t) = V_0 \sin(\omega t)$$
From the above equation, Vin in s-domain is then:
$$V_{in}(s)=V_0\frac{\omega}{s^2+\omega^2}$$
And Vout in s-domain is as follows:
$$V_{out}(s)=V_{in}(s)\times \frac{\frac{-1}{C_1s}}{R_1}=V_{in}(s)\times\frac{-1}{R_1C_1s}$$
Subsitituting Vin(s) in the equation above, we get:
$$V_{out}(s)=\frac{-V_0}{R_1C_1}\frac{1}{s}\frac{\omega}{s^2+\omega^2}$$
With partial fraction, it becomes:
$$V_{out}(s)=\frac{-V_0}{R_1C_1}[\frac{1\over\omega}{s}+\frac{-s\over\omega}{s^2+\omega^2}] =\frac{-V_0}{R_1C_1\omega}[\frac{1}{s}+\frac{-s}{s^2+\omega^2}]$$
Convert back to time domain, it becomes:
$$V_{out}(t)=\frac{-V_0}{R_1C_1\omega}[1-\cos(\omega t)]=\frac{V_0 \cos(\omega t)}{R_1C_1\omega}-\frac{V_0}{R_1C_1\omega}$$
The answer key from the book is:
$$\frac{V_0 \cos(\omega t)}{R_1C_1\omega}$$
Where did I do wrong?
simulate this circuit – Schematic created using CircuitLab
