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a. The function x3 is the simplest function whose derivative is 3x2, so
b. is the collection of all functions with derivative ex. The function ex is its own derivative and antiderivative. This means any antiderivative of ex looks like ex plus some constant.
c. Every antiderivative of 0 is a constant, so
We know that is the family of all antiderivatives of exsin(ex). This means if we take the derivative of our answer, -cos(ex) + C, we should get exsin(ex) back again. Let's see if that happens. We won't bother taking the derivative of C since we know we'll get 0.