1. 1. Antiderivatives and Indefinite Integration

2. Antiderivatives
Suppose we have a function F whose derivative is given as
From your experience in finding derivatives, you could say that since
By using your intuition, you have just found an antiderivative.
Notice I said AN antiderivative, not THE antiderivative
What are some other possible antiderivatives of this function?

3. General Antiderivative
Because of this, we can say that the general antiderivative of a function f(x) is F(x) + C, where F is the antiderivative and C is an arbitrary constant.
The graph below shows several members of the family of the antiderivatives of x2.

4. Example 1
Find the general antiderivatives using your knowledge of derivatives.

5. Some antiderivative formulas

6. Example 2
Find the general antiderivative of

7. Differential Equation
A differential equation is an equation explicitly solved for a derivative of a particular equation.
Solving a differential equation involves finding the original function from which the derivative came.
The general solution involves +C.
The particular solution uses an initial condition to find the specific value of C.

8. Example 3
Solve the differential equation if f(2)= -3. Find both the general and particular solutions.

9. Example 4
Find the particular solution to and y(0)=-2

10. Example 5 Find the particular solution to and A) y’(1)=3 and y(0)=-6
B) y’(0)=4 and y(1)=1

11. Indefinite integration
The process of finding antiderivatives is called indefinite integration. We can denote this operation with an integral symbol.
Here is the whole notation
Integrand
Integral symbol
Variable of Integration

12. Example 6