A. An
integral is basically the exact opposite of a *derivative*.

B. An
integral gives the area under a curve from the x-axis to the curve from a
specified range. An integral is expressed by the symbol: _{}.

C. Below are some basic rules for computing integrals:

1. The integral of a constant, c, is cx.

2. The
integral of x^{n} is _{}.

3. The integral of a*f(x), where a is a constant, is equal to a times the integral of f(x). (*Basically, you can factor out the constant*)

4. The
integral of f(x) _{} g(x) = _{}.

a. In this problem, we can take each term separate.

b. The
integral of 3x^{2 }= x^{3}; of 2x = x^{2}; of 1 = x.

c. The
answer is x^{3}+x^{2}+x+C

d. We usually add a C on the end because the integral of 0 is a constant.

D. You should notice, that if you take the derivative of the new function, you will end up with the original function.

E. Common Integrals:

1. The
integral of sin(ax) = -^{1}/_{a} cos(ax)

2. The
integral of cos(ax) = ^{1}/_{a} sin(ax)

3. The
integral of sec^{2} x = tan (x)

4. The
integral of e^{(ax)} = ^{1}/_{a} e^{(ax)}.

5. The
integral of ^{1}/_{ax} = ^{1}/_{a }ln(ax)

F. These are just some of the common integrals. There are lots of integrals in many different forms, which can be found on the front or back cover of most calculus books.

G. When calculating an integral in a specified range:

a. To
compute this integral take each term separate and get: x^{3}/_{3}
- x^{2}.

b. To find
the value from 0 to 3, first plug in 3. Doing so gives: 3^{3}/_{3}
- 3^{2} = 9 - 9 = 0.

c. Next find
the value for 0. This gives: 0^{3}/_{3} - 0^{2} =
0.

d. The answer is 0 - 0 = 0.

a. The
integral of cos 4x is ^{sin 4x}/_{4}.

b. To find
the value from 0 to _{} we
must first use _{}.
Doing so gives: ^{sin 3(/2)}/_{3}
= ^{sin(3/2)}/_{3 }= -^{1}/_{3}.

c. Next,
find the value for 0. This is ^{sin 0}/_{4} = 0.

d. The
answer is -^{1}/_{3} - 0 = -^{1}/_{3}.

a. The
integral of x^{2} - 2 is x^{3}/_{3} - 2x.

b. Plug in 4
and get ^{64}/_{3} - 8 = ^{40}/_{3}.

c. Plug in 1
and get ^{1}/_{3} - 2 = -^{5}/_{3}.

d. The
answer is ^{40}/_{3} - (-^{5}/_{3}) = ^{45}/_{3}
= 15.