This document discusses indefinite integrals and anti-derivatives. It explains that the indefinite integral of a function is the set of all anti-derivatives of that function. It provides rules for finding the anti-derivative of a function given its derivative, such as adding 1 to the power and dividing by the new power. Examples are given of using these rules to find the function given its derivative.

1. Anti-derivatives.
Indefinite Integrals
By the end of this lesson you will be able to:
• Understand the integration process as the
reverse process of differentiation.
• Calculate the integrals of powers of x and any
linear combination of them.

2. f is a function whose derivative is f '(x) = 2 x
then f(x) =
or f(x) =
or f(x) =
in general f(x) =
f is the antiderivative of f '

3. The set of all the antiderivatives of a function is
called the indefinite integral of the function
∫
So:
Note: Sometimes we write: F(x) to mean the integral of f(x). So we
have
F(x) = ∫ f(x) dx

4. f(x)
f '(x)
x
x2
x3
x
n
1) Multiply by the power
2) Subtract 1 to the power
f '(x)
f(x)
1
x
x2
n
x
1) Add 1 to the power
2) Divide by the power

5. In general:

11. (
(

15. The gradient function of a function H is h(x) = x +1
What is the function H?
If the graph of the function H passes through (1,4),
give the equation of H.

16. A particular curve has
We are told that the curve passes through the point (2, 5).
Find the equation of the curve.

17. differentiation
s = displacement
integration

18. The acceleration a ms­2 of a particle moving in a
straight line is given by a = t +1. Find formulae
for the velocity and the displacement if s =0 and
v=8 when t=0.

19. Attachments
Integral=Sum.swf