Well tell us what you do know about for this problem. For instance what do you know about getting random samples from arbitrary distributions? Do you know what a Markov Chain random sampler is or how to program one? If you had a random sample from that distribution would you know how to compute the monte carlo estimate of the integral?
As long as you make sure that the density of that gamma is always greater than the density you're trying to sample from then you're ok. You'll need to prove to yourself at least that this is the case. (But you might even want to multiply the desired density by a constant so that you aren't always rejecting your proposed point).
I would choose a density function which is always greater than the density you're trying to sample from. The term exp(-3x)*(1- exp(-2x))^7 is always smaller than 1. So, you can take g(x) = x^4. I am not sure. What do you think, Dason?
Ok, to apply the rejection method it must be a function whose indefinite integral is known analytically and is also analytically invertible (see Numerical Recipes, Press et al. p. 295). I plotted the function from 0 to 10 and integrated it. The proportional constant is 15.42493. Then the area under the density function is 1. There is only very few contribution to the area, if x is greater 10. So, I thought to regard only the interval 0 to 10.
So, you can take g(x) = a*x^4*exp(-3x) as upper function. It's always greater than f(x). But you also need rejection sampling for the gamma density. The comparison function is then of the form h(x) = c0/(1 + (x - x0)^2/a0^2). Then you can sample x = a0 * tan(pi*U) + x0 where U is a uniform deviate between 0 and 1. You must choose the values of a0, b0 and c0 such that h(x) is everywhere greater than f(x). I read this in Numerical Recipes on p.296.
Another possibility is that you can add up 5 i.i.d exponential(3) random variates to obtain one gamma (though it may not be very efficient)
The one fanky points out should be some how like using the Cauchy random variate as the proposal density. Anyway the main point of the proposal distribution is that it should have the similar shape with the target to increase the efficiency, and also it should be easy to simulate.
The one I am most interested here is the Markov Chain sampler though