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The concepts of radioactive half-life and mean life play crucial roles in studying radioactive isotopes and their decay. The half-life (\(T_{1/2}\)) tells us the time required for half of the radioactive atoms in a sample to decay, but it doesn't provide information about when specific atoms will decay. Here is where the concept of mean life, denoted as \(\tau\), becomes significant. Mean life represents the average lifespan of all the atoms before they decay, providing a broader statistical view.

To illustrate, imagine a large group of atoms; each has its clock ticking to an unpredictable decay moment. You cannot tell exactly when each will 'chime,' but you can estimate an average 'chime time' for the entire group—that's the mean life. Now, if we think of the decay of atoms as a natural 'clock,' the mean life is akin to an average time each of these clocks runs before stopping.

The mathematical expression to calculate the mean life is \(\tau = 1/\lambda\), where \(\lambda\) is the decay constant—a rate that quantifies how quickly atoms of a radioactive isotope undergo decay. In essence, the higher the decay constant, the shorter the mean life, indicating a faster decaying substance.

To grasp the decay law for radioactive substances, one must understand their probabilistic nature. The decay law mathematically describes how the quantity of undecayed atoms decreases over time. It is captured by the equation \(N(t) = N_0e^{-\lambda t}\), where \(N(t)\) is the number of undecayed atoms at time \(t\), \(N_0\) is the initial number, and \(\lambda\) is the decay constant.

For a more in-depth insight, consider the probability density function \(f(t) = \lambda e^{-\lambda t}\). This function represents the likelihood of an atom decaying at time \(t\). The integral of this function over time gives us mean life as it accounts for all possible 'decay times' weighted by their probability of occurrence.

By delving into the interplay between the decay law and the concept of mean life, one comprehends that the inevitable randomness of radioactive decay is not so chaotic after all—it adheres to a predictable pattern over a large sample and extended time, governed by this prolific decay law.

The technique of integration by parts is a powerful tool in calculus, especially when working with products of functions. It serves as the mathematical bridge connecting the decay law and mean life of a radioactive isotope. The formula for integration by parts is given by \(\int u \(\text{d}v\) = uv - \int v \(\text{d}u\)\).

For our problem, we set \(u = t\) and \(dv = \lambda e^{-\lambda t} \(\text{d}t\)\), resulting in the integral \(\int_0^{\infty} t \lambda e^{-\lambda t} \(\text{d}t\)\). Using integration by parts, we find not just an abstract formula but a concrete number that encapsulates the essence of a radioactive sample's behavior over time—the mean life.

Through this computation, we get \(\tau = 1/\lambda\), a surprisingly simple yet profoundly meaningful outcome. This process illuminates how a seemingly challenging calculation can be broken down and decoded using a clever algebraic methodology.