If a direct current of value a ampere is superimposed on an alternative current$I=b \sin \omega t$ flowing through a wire, what is the effective value of the resulting current in the circuit?

As current at any instant in the circuit will be

$I=I_{dc}+I_{ac}=a+b \sin \omega t$

$so, I_{eff}=\left[\frac{{\displaystyle {\int }_0^T{I}^{2}dt}}{{\displaystyle {\int }_0^{T}dt}}\right]={\left[\frac{{\displaystyle 1}}{{\displaystyle T}}{\int }_0^T{(a+bsin\omega t)}^{2}dt\right]}^{1/2}$

$i.e., I_{eff}={\left[\frac{{\displaystyle 1}}{{\displaystyle T}}{\int }_0^T(a^2+2absin\omega t+b^2{\sin }^2\omega t)dt\right]}^{1/2}$

 $but as \frac{1}{T}{\int }_0^{T}sin\omega t dt=0 and \frac{1}{T}{\int }_0^T{sin}^2 \omega t dt=\frac{1}{2}$

$So, I_{eff}={\left[a^2+\frac{1}{2}{b}^2\right]}^{1/2}$