Solving double (n-tuple) multivariate integrals in R

I would like to write a code to solve this kind of equations:

For that I wrote the code below, however it does not solve the problem. Do you have any ideas about the possibility to solve this kind of integrals in R?

t_0 = 15
mu = 0.1
lambda = 0.8
f = function(x1,x2) exp(mu*(x1+x2))*dexp(log(lambda)*(x1+x2))
f_comp = function(x2) f(x1,x2)
f_1 = function(x1) {integrate(f_comp,upper = t_0, lower = x1)}
result = integrate(f = f_1, lower = 0, upper = t_0)$value

--------- edit:

Given the answer below, I adapt the code to my example, but I still think is not the correct one, at least the value 0 for the integral does not make sense.

integrate(function(x1) {
  sapply(x1, function(x1){
    integrate(function(x2)  exp(mu*(x1+x2))*dexp(log(lambda)*(x1+x2)), lower = x1, upper = t_0)$value
  })
}, 0, t_0)

by the way, I would like to buid a general procedure for that (that is why I just not calculate the integral by hand). That is not only double integrals, but also n-tuples integrals, so I need a general procedure for this kind of calculations.