you have the right approach
Lets find P(Y<y)
Find (or look up) F(x) then subsititute these values in to get the CDF of Y. Hopefully it will look like an exponential distribution.
is a random variable with weibull distribution given with
Let and .
Show that has exponential distribution and find value of parameter .
I believe that should be expressed in terms of , but I don't know where to start.
But if you're determined to use a distribution function approach, then you should start by noting that the cdf of Y is:
since the support of X is .
Now set up the required integral (no need to integrate by the way ....) and then differentiate using the chain rule and the Fundamental Theorem of Calculus.