For AP Calculus Class; 8.1 ALL Basic Integration Rules

1. All Basic Integration Rules
Explained by Kaitlin Bianchi

2. Basic Integration
• The symbol ∫ • An integral is
refers to 'the basically the
integral of' or more exact opposite
commonly 'the
area under’ from
of a derivative.
the x-axis to the
curve from a
specified range.

3. The next few slides are some
basic rules for computing
integrals

4. Rule 1: The integral of a
constant, c, is cx
Example:
∫5dx = 5x

5. Rule 2: The integral of xn is
xn+1/n+1
Example:
∫x5dx = x6/6

6. Rule 3: The integral of a*f(x), where a
is a constant, is equal to a times the
integral of f(x). (In simpler terms this
means you can factor out the
constant).
Example:
∫3xdx = 3 ∫xdx = 3*x2/2 = 3x2/2

7. Rule 4: The integral of
f(x) + g(x) = ∫f(x) + ∫g(x)
Example:
∫(3x2 + 2x + 1)dx =
In this problem, we can take each term separate.
The integral of 3x2 = x3 ; of 2x = x2 ; of 1 = x
The answer is x3 + x2 + x + C
Note: We usually add a C on the end because
the integral of 0 is a constant.

8. If ever you want to check
your answer to make sure
that it is correct, remember
that if you take the derivative
of the new function, you
should end up with the
original function.

9. The following are some other
common integrals:
1) ∫ sin(ax) = -1/a cos(ax) 2)
∫ cos(ax) = 1/a sin(ax) 3) ∫
sec2 x = tan (x)
4) The integral of e(ax) = 1/a e(ax).
5. The integral of 1/ax= 1/a ln(ax)

10. When calculating an integral in a
specified range:
a ∫b f(x)dx = ∫f(b) - ∫f(a)
Example:
0 ∫3 (x2 - 2x)dx
To solve this integral take each term separate and get:
x3/3 - x2
To find the value from 0 to 3, first plug in 3. Doing so
gives 33/3 - 32 = 9 - 9 = 0
Next find the value for 0. This gives 03/3 - 02 = 0
The answer is 0 - 0 = 0

11. Practice
Problems

12. Practice Problem #1
∫5e3xdx
5e3x/3 + C

13. Practice Problem #2
∫sec(2x)tan(2x)dx
=1/2 sec(2x) + C

14. Practice Problem #3
∫2/(3x + 1)dx
= 2/3 ln I 3x + 1 I + C

15. Practice Problem #4
0∫π/2 cos3xdx
The integral of cos3x is sin3x/3
To find the value from 0 to π/2
we must first plug in π/2 for x.
Doing so gives: sin3(π/2)/3
= -1/3

16. Practice Problem #5
1∫4 (x2 - 2)dx
The integral of x2-2 is x3/3 - 2x
You then plug in 4 and get 64/3 -8
= 40/3.
You then plug in 1 and get
1/3 - 2 = -5/3
The answer is 40/3 - (-5/3) = 45/3
= 15

17. This ends the presentation. I
hope this helped your
understanding of basic
integration rules.

18. "Mathematics compares the most
diverse phenomena and
discovers the secret analogies
that unite them." Jean Baptiste
Joseph Fourier (1768-1830)