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a. The function x^{3} is the simplest function whose derivative is 3x^{2}, so

b. is the collection of all functions with derivative e^{x}. The function e^{x} is its own derivative and antiderivative. This means any antiderivative of e^{x} looks like e^{x} plus some constant.

c. Every antiderivative of 0 is a constant, so

Example 2

Does

We know that is the family of all antiderivatives of e^{x}sin(e^{x}). This means if we take the derivative of our answer, -cos(e^{x}) + C, we should get e^{x}sin(e^{x}) back again. Let's see if that happens. We won't bother taking the derivative of C since we know we'll get 0.