If a direct current of  “a” ampere is super imposed on an alternating current I=bsinωt flowing through a wire, what is the effective value of resulting current in the circuit

GivenDirect current  Idc=aAlternative current  Iac=bsinωtTotal currentI=Idc+Iac⇒I=a+bsin⁡ωt⇒Irms2=∫i2dt∫dt=1T∫0T (a+bsin⁡ωt)2dt⇒Irms2=1T∫0T a2dt+∫oT b2sin2⁡ωt+∫0T 2basin⁡ωtWe know, ∫0T sin⁡ωtdt=0 and∫0T sin2⁡ωtdt=T2⇒Irms2=1Ta2T+b2T2+0∴Irms=a2+b221/2