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Preview Activity 5.4.1. In Section 2.3, we developed the Product Rule and studied how it is employed to differentiate a product of two functions. In particular, recall that iff and g are differentiable function of d r x, thenE [f(x} - g(x)] = f(x) - 91x) + 9(x) - f (x). a. For each of the following functions, use the Product Rule to find the function's derivative. Be sure to label each derivative by name (e.g., the derivative of 30:) should be labeled 3'00}. i. 90:) = x sin(x} Iv. (30:) = x2 cos(x) ii. h(x) = 178" v. r'(x) = e'sinc) iii. 110:) = xln[x) b. Use your work in (a) to help you evaluate the following indenite integrals. Use differentiation to check your work. i. fxex + 8" dx iv. Ix cos[x) + sin(x) dx ii. I ex(sin(x) + cos(x)) dx v. I 1 + 1110:) dx iii. f2x cos(x) x2 sin(x) dx c. Observe that the examples in [bi work nicely because of the derivatives you were asked to calculate in [a]. Each integrand in {b} is precisely the result of differentiating one of the products of basic functions found in {a}. To see what happens when an integrand is still a product but not necessarily the result of differentiating an elementary product, we consider how to evaluate Ix (2050:) dx. i. First observe that % [x sin(x)] = x cos(x) + sirl(x). Integrating both sides indenitely and using the fact that the integral of a sum is the sum of the integrals, we find that f (Cf1L1: sin(x)]) dx = Ix cos(x) dx + f sin(x} dx. In this last equation, evaluate the indenite integral on the left side as well as the rightmost indenite integral on the right. ii. In the most recent equation from {i}, solve the equation for the expression Ix cos(x) dx. iii. For which product of basic functions have you now found the antiderivative