Final answer:
The question involves determining the convolution of functions using direct integration, a process integral to mathematics and signal processing. Without complete information, convolutions for the given pairs cannot be achieved; however, an example of convolution for unit step functions is provided.
Step-by-step explanation:
The question requires the determination of the convolution of a set of functions using direct integration. This involves integrating the product of the functions over a specified range, taking into the account the characteristics of the functions involved, such as step functions u(t) or ramp functions r(t). Importantly, the convolution of signals is a fundamental concept in mathematics and engineering that combines two signals to form a third signal that expresses how the shape of one is modified by the other.
Without the proper limits and functions, a convolution can't be found. The provided snippets of information are incomplete and can't be used to complete the convolutions (a) u(t) * u(t), (b) r(t) * u(t), (c) sin(t)u(t) * u(t), and (d) 2 sin(htt) u(t) * (8(t) + 8(t – 1)).
To illustrate, the convolution of u(t) * u(t), for example, would involve integrating from -∞ to +∞ the product of two unit step functions which becomes an integration from 0 to t, leading to a result of t for t ≥0.