The main difference between these two functions is that sinh(x) * sin(x) is a product of two hyperbolic sine functions, while sinh(x) * cos(x) is a product of a hyperbolic sine and a hyperbolic cosine function. This results in different graphs and behaviors, as well as different integral and derivative formulas.
The graph of sinh(x) * sin(x) is a symmetric, oscillating curve that approaches zero as x approaches infinity. It has a maximum value of 1 at x = 0 and a minimum value of -1 at x = π.
To differentiate sinh(x) * cos(x), you can use the product rule of differentiation, which states that the derivative of a product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function. In this case, the derivative is cosh(x) * cos(x) - sinh(x) * sin(x).
The integral of sinh(x) * sin(x) can be found by using the integration by parts method, which states that the integral of a product of two functions is equal to the first function multiplied by the integral of the second function, minus the integral of the derivative of the first function multiplied by the integral of the second function. In this case, the integral is (sinh(x) * cos(x)) / 2 - (cosh(x) * sin(x)) / 2 + C, where C is the constant of integration.
These functions are commonly used in fields such as physics and engineering to model various natural phenomena. For example, in electrical engineering, they are used to describe the behavior of alternating current in circuits. In addition, they are also used in statistics and probability to model random processes and distributions.