derive an expression for average life of a decaying nucleus.

Average life of a decaying nucleus is also known as the mean life , τ. Let the activity be R . Then the number of nuclei decayed during the very small time interval dt is given by Rdt = Nλdt = N₀e^-λ​t λ​ .
All these nuclei have lived for a time t.
The total life of these nuclei = N₀e^-λ​t λ​ t dt​,
But clearly , some nuclei will live more or less life than these . In order to get mean life , we need to take the sum of lifes of all particles and we need to divide it  by total number of particles ( N₀ ).
So mean life = Life of Particle 1 + Life of particle 2 + .........../N₀
This sum involves infinitely many terms . So it is an integral . 
 I will show you  how to integrate this  but it is not really necessary . Even for the exam  , you could write up to the first step in the integral . Then you can write " By performing this integral  , we get " and directly write the final answer . ( If you are not a maths student , you could write like this. But if you are a math student and like to see how this is done  , please see the attached figure )

so τ = 1/λ.
Hope that helps.