The integrated rate equation for first order reaction, $A\to$ product, is

The integrated rate equation for first order
Reaction, $A\to$ product is $k=\frac{1}{t}In\frac{{\left[A\right]}_t}{{\left[A\right]}_0}.$
For 1^st order reactions, $R\to P$
Rate = $\frac{-d\left[R\right]}{dt}=\stackrel{·}{k}\left[R\right]$
On integrating this equation, we get
In $\left[R\right]=-kt+l...\left(i\right)$
At $t=0,R={\left[R\right]}_0=-k\times 0+l$
In ${\left[R\right]}_0=l$
Substituting the value of l in (i) and on rearrangement, we get
In $\left[R\right]=-kt+In{R}_0$
And, $k=\frac{1}{t}In\frac{\left[R_0\right]}{\left[R\right]}$