# Integral of survival function

If X is a nonnegative random variable representing the life of a component having distribution function F, and S is the survival function and T be a nonnegative random variable representing the time until some specied event. What is the result of this integral?

$$\int_0^x \int_0^t S(u) du dt + \int_x^\infty \int_t^\infty S(u) du dt$$

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 What is u in this context? Have you already made a substitution to facilitate integration? If so, please include it in the question. Is u supposed be x? If so, please change it. Is u an additional variable that is a function of x? If so, please define it. – John Doucette Oct 9 '12 at 12:14 It looks like u is just a placeholder variable. It goes away after taking the first integral. – wcampbell Oct 9 '12 at 14:01 u is a placeholder variable. – Bensor Beny Oct 9 '12 at 22:31 If this is homework, please add the corresponding tag. What have you tried? – Zen Oct 10 '12 at 0:53 It's not a homework. I would like to know whats i sthe result of first integral and the second one. because i do not know what is the integral of survival function in these partitions. – Bensor Beny Oct 10 '12 at 2:33