Indefinite Integrals Introduction Examples

Evaluate each indefinite integral.

a.

b.

c.

a. The function x³ is the simplest function whose derivative is 3x², 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

Does

We know that is the family of all antiderivatives of e^xsin(e^x). This means if we take the derivative of our answer, -cos(e^x) + C, we should get e^xsin(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.

Yup, our answer was right!