If a direct current of value a ampere is superimposed on an alternative currentI=b sinωt 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 beI=Idc+Iac=a+b sin ωtso, Ieff=∫0TI2dt∫0Tdt=1T∫0T(a+bsinωt)2dt1/2i.e., Ieff=1T∫0T(a2+2absinωt+b2sin2ωt)dt1/2 but as 1T∫0Tsinωt dt=0  and  1T∫0Tsin2 ωt dt=12So, Ieff=a2+12b21/2