Let $Q(t)$ be the charge stored in a capacitor $C$ in $t$ time for alternating current. Now, $$\displaystyle Q(t)=Q(0)+\int_{0}^{t}I(t)dt\\=Q(0)+\int_{0}^{t}I_oe^{j\omega t}dt\\=Q(0)+\frac{I_oe^{j\omega t}-I_o}{j\omega}\\=Q(0)+\frac{I(t)-I_o}{j\omega}$$

It yields, $\displaystyle C=\frac{Q(t)}{V(t)}= \frac{Q(0)}{V(t)}+\frac{I(t)}{j\omega V(t)}-\frac{I_o}{j\omega V(t)}$

But, reactance of capacitor, $\displaystyle X_c=\frac{V(t)}{I(t)}=\frac{1}{j\omega C}$. Constant term $\left(\frac{I_o}{j\omega}\right)$ of the integration seems to be unexpected here. What's wrong in this approach?

Let $Q(t)$ be the charge stored in a capacitor $C$ in $t$ time for alternating current. Now, $$\displaystyle Q(t)=Q(0)+\int_{0}^{t}I(t)dt\\=Q(0)+\int_{0}^{t}I_oe^{j\omega t}dt\\=Q(0)+\frac{I_oe^{j\omega t}-I_o}{j\omega}\\=Q(0)+\frac{I(t)-I_o}{j\omega}$$

It yields, $$\displaystyle C=\frac{Q(t)}{V(t)}= \frac{Q(0)}{V(t)}+\frac{I(t)}{j\omega V(t)}-\frac{I_o}{j\omega V(t)}.$$

But, reactance of capacitor, $$\displaystyle X_c=\frac{V(t)}{I(t)}=\frac{1}{j\omega C}.$$ Constant term $\left(\frac{I_o}{j\omega}\right)$ of the integration seems to be unexpected here. What's wrong in this approach?

Let $Q(t)$ be the charge stored in a capacitor $C$ in $t$ time for alternating current. Now, $\displaystyle Q(t)=Q(0)+\int_{0}^{t}I(t)dt\\=Q(0)+\int_{0}^{t}I_oe^{j\omega t}dt\\=Q(0)+\frac{I_oe^{j\omega t}-I_o}{j\omega}\\=Q(0)+\frac{I(t)-I_o}{j\omega}$

It yields, $\displaystyle C=\frac{Q(t)}{V(t)}= \frac{Q(0)}{V(t)}+\frac{I(t)}{j\omega V(t)}-\frac{I_o}{j\omega V(t)}$

But, reactance of capacitor, $\displaystyle X_c=\frac{V(t)}{I(t)}=\frac{1}{j\omega C}$. Constant term ($\frac{I_o}{j\omega}$) of the integration seems to be unexpected here. What's wrong in this approach?

Let $Q(t)$ be the charge stored in a capacitor $C$ in $t$ time for alternating current. Now, $$\displaystyle Q(t)=Q(0)+\int_{0}^{t}I(t)dt\\=Q(0)+\int_{0}^{t}I_oe^{j\omega t}dt\\=Q(0)+\frac{I_oe^{j\omega t}-I_o}{j\omega}\\=Q(0)+\frac{I(t)-I_o}{j\omega}$$

It yields, $\displaystyle C=\frac{Q(t)}{V(t)}= \frac{Q(0)}{V(t)}+\frac{I(t)}{j\omega V(t)}-\frac{I_o}{j\omega V(t)}$

But, reactance of capacitor, $\displaystyle X_c=\frac{V(t)}{I(t)}=\frac{1}{j\omega C}$. Constant term $\left(\frac{I_o}{j\omega}\right)$ of the integration seems to be unexpected here. What's wrong in this approach?

Let $Q(t)$ be the charge stored in a capacitor $C$ in $t$ time for alternating current. Now, $\displaystyle Q(t)=\int_{0}^{t}I(t)dt=\int_{0}^{t}I_oe^{j\omega t}dt=\frac{I_oe^{j\omega t}-I_o}{j\omega}=\frac{I(t)-I_o}{j\omega}$

It yields, $\displaystyle C=\frac{Q(t)}{V(t)}= \frac{I(t)}{j\omega V(t)}-\frac{I_o}{j\omega V(t)}$

But, reactance of capacitor, $\displaystyle X_c=\frac{V(t)}{I(t)}=\frac{1}{j\omega C}$. Constant term ($\frac{I_o}{j\omega}$) of the integration seems to be unexpected here. What's wrong in this approach?

Let $Q(t)$ be the charge stored in a capacitor $C$ in $t$ time for alternating current. Now, $\displaystyle Q(t)=Q(0)+\int_{0}^{t}I(t)dt\\=Q(0)+\int_{0}^{t}I_oe^{j\omega t}dt\\=Q(0)+\frac{I_oe^{j\omega t}-I_o}{j\omega}\\=Q(0)+\frac{I(t)-I_o}{j\omega}$

It yields, $\displaystyle C=\frac{Q(t)}{V(t)}= \frac{Q(0)}{V(t)}+\frac{I(t)}{j\omega V(t)}-\frac{I_o}{j\omega V(t)}$

But, reactance of capacitor, $\displaystyle X_c=\frac{V(t)}{I(t)}=\frac{1}{j\omega C}$. Constant term ($\frac{I_o}{j\omega}$) of the integration seems to be unexpected here. What's wrong in this approach?