State radioactive decay law. Derive `N=N_0e^(-lambdat)` for a radioactive element

Explanation

The number of atoms disintegrated per second at any instant is directly proportional to the number of radioactive atoms actually present in the sample at the instant.
Mathematically , `(dN)/(dt)=N`
Here N is the number of radioactive nuclei present in the sample at an instant.
`(dN)/(dt)=-lambdaN`
where `lambda` is disintegration constant.
`(dN)/(dt)=-lambdadt`
`int(dN)/N=-lambda int dt`
At t=0, `N=N_0 " " log_e N=-lambdat+C`
i.e., `C=log_e N_0`
Thus, `log_e N=-lambdat+log_e N_0`
`log_e (N/N_0)=-lambdat`
As `N=N_0 e^(-lambdat)`
Alternate correct method should be considered.