Dear student,

We will solve this integration using ' By Parts '.

Now take a^x as the 1st function and e^x as the 2nd function, according to the ILATE Rule.

Now We use, intg(1*2) = 1.intg(2) - intg.{diff.(1)intg(2)}

Let, I be the answer of this integration then,

I = a^x.e^x - intg[a^x. log(a). e^x] now we can write it as

I = a^x.e^x - log(a). I

so clearly I = a^x.e^x/ 1+log(a)

Hope this helps.