1. Antidifferentiation and Indefinite Integrals

2. A function F is an antiderivative of f on an interval I if for all x in I.
Be careful though. If we are looking for a function whose derivative is
a lot of different answers can be found.
The functions:
are all correct antiderivatives.

3. Representation of Antiderivatives
F is an antiderivative of f if:
1. C is called the constant of integration
2. F(x) + C is called the general solution of the differential equation
3. y = f ‘(x) is the differential equation.

4. Find the general solution of .
Answer:
The operation of finding all solutions of a differential equation is called antidifferentiation or indefinite integration.
To show antidifferentiation, we use the symbol:

5. Variable of Integration
Constant of Integration
Integrand
is the antiderivative of f with respect to x

6. Basic Rules of Integration
1. Integration is the inverse of differentiation.
2. Differentiation is the inverse of integration.
Integration Using the Power Rule

7. Find:
Answer:
Find:
Answer:
Find:
Answer:

8. Find:
Answer:
Answer:
Find:

9. Find:
Answer:
Find:
Answer:

10. Suppose that the derivative of y is given below and that y = -50 when x = 3. Find y.
Answer:
Suppose that the derivative of y is given below and that y = 2 when x = 1. Find y.
Answer:

11. Suppose the derivative of y is Find y.
Answer:
Suppose that the derivative of y is given below and that y = 8 when x = 1. Find y.
Answer:

12. Second Order Differentials
If the derivative of a derivative is written as: or
then:
gives us the antiderivative or indefinite integral of a second derivative.

13. Find y as a function of x, given that and that when x = 2, and y = 20.
Answer:
Find y as a function of x, given that and that y = 4 when x = 1 and y = 2 when x = -1.
Answer:

14. The gradient of a curve at the point (x, y) on the curve is given by
The gradient of a curve at the point (x, y) on the curve is given by If the curve and the line 2x - y - 1 = 0 intersect the y-axis at the same point, find the equation of the curve.
Answer:
The gradient of a curve at the point (x, y) on the curve is given by (2x – 4). If the minimum value of y is 3, find the equation of the curve.
Answer: