Nuclei of a radioactive element are being produced at a constant rate α. The element has a decay constant λ. At time t = 0, there are N0 nuclei of the element.
The number N of nuclei at time t is
To calculate the number N of nuclei at time t (given that the number at t = 0 is N0), we integrate the above expression from N = N0 to N = N and from t = 0 to t = t, We have
or loge(α−λNα−λN0)=−λt or α−λNα−λN0=e−λt