If a direct current of  “a” ampere is super imposed on an alternating current $I=b\sin \omega t$ flowing through a wire, what is the effective value of resulting current in the circuit

Given
Direct current  $I_{dc}=a$
Alternative current  $I_{ac}=b\sin \omega t$
Total current
$\begin{array}{l}I=I_{dc}+I_{ac}\\ \Rightarrow I=a+b\sin \omega t\\ \Rightarrow I_{\mathrm{rms}}^2=\frac{\int i^2\mathrm{d}t}{\int \mathrm{d}t}=\frac{1}{T}{\int }_0^T (a+b\sin \omega t{)}^2\mathrm{d}t\\ \Rightarrow I_{\mathrm{rms}}^2=\frac{1}{T}\left[{\int }_0^T a^2\mathrm{d}t+{\int }_o^T b^2{\sin }^2\omega t dt+{\int }_0^T 2ba\sin \omega t dt\right]\\ We know, {\int }_0^T \sin \omega t\mathrm{d}t=0 and\\ {\int }_0^T {\sin }^2\omega t\mathrm{d}t=\frac{T}{2}\end{array}$

$\begin{array}{l}\Rightarrow I_{rms}^2=\frac{1}{T}\left[a^{2}T+b^2\frac{T}{2}+0\right]\\ \therefore I_{\mathrm{rms}}={\left[a^2+\frac{b^2}{2}\right]}^{1/2}\end{array}$