Power Rule

A presentation of the topic on power rule

1. Power Formulas
Joey F. Valdriz

2. 2
A. Power of a Variable The integral of a variable to a power is the variable to a power increased by one and
divided by the new power.

Formula:

3. Examples
3 EXAMPLE: Evaluate

SOLUTION:

EXAMPLE: Evaluate

SOLUTION:

4. B. Constants
4 A constant may be written either before or after the integral sign.

Formula:

5. Examples
5 Example: Evaluate

Solution:

Example: Evaluate

Solution:

6. C. Sums
6 The integral of an algebraic sum of differentiable functions is the same as the
algebraic sum of the integrals of these functions taken separately; that is, the
integral of a sum is the sum of the integrals.

Formula:

7. Examples
7 Example: Evaluate

Solution:


Example: Evaluate

Solution:

8. D. Power of a Function
8 The integral of a function raised to a power is found by the following steps:
1. Increase the power of the function by 1.
2. Divide the result of step 1 by this increased power. 3. Add the constant of
integration.

Formula:

9. Examples
9 Example: Evaluate

Solution: Let so that or
Then by substitution,

Therefore,

10. 10 Example: Evaluate

Solution: Let so that

We find dx in the integral but not 3 dx. A 3 must be included in the integral to fulfill
the requirements of du. In words, this means the integral needs du so
that the formula may be used. Therefore, we write and recalling that
a constant may be carried across the integral sign, we write

Notice that we needed 3 in the integral for du, and we included 3 in the integral;
we then compensated for the 3 by multiplying the integral by 1/3.
Then

11. 11 Example: Evaluate

Solution: Let so that

Then

12. E. Quotient
(Method 1: Putting the quotient into the form of the power of a
function)
12

13. Examples
13 Example: Evaluate

Solution:

Let so that

The factor 2 is used in the integral to give du and is compensated for by multiplying the
integral by 1/2.
Therefore,

14. 14
E. Quotient
(Method 2: Operations with logarithms) In the previous formulas for integration of a function, the exponent was not allowed to
be -1. In the special case of where
we would have applied the following formula:

Formula:

15. 15
E. Quotient
(Method 3: A special case in which the quotient must be
simplified to use the sum rule) In the third method for solving integrals of quotients, we find that to integrate
an algebraic function with a numerator that is not of lower degree than its
denominator, we proceed as follows: Change the integrand into a polynomial plus a
fraction by dividing the denominator into the numerator. After this is
accomplished, apply the rules available.

16. 16 Example: Evaluate

Solution:
Divide the denominator into the numerator so that

Integrating each term separately, we have
and and

Then, by substitution, we find that

Where

17. 17 Example: Evaluate

Solution:
The numerator in not of lower degree than the denominator; therefore, we
divide and find that

Integrating each term separately, we find that and


Therefore,
where

18. Recall
Previous learning:
If f(x) = e
x
then f’(x) = e
x

If f(x) = a
x
then f’(x) = (ln a)a
x

If f(x) = e
kx
then f’(x) = ke
kx

If f(x) = a
kx
then f’(x) = k(ln a)a
kx


This leads to the following formulas:

19. Indefinite Integrals of
Exponential Functions
19 ln
kx
kx a
a dx C
ka

 ln
x
x a
a dx C
a
 kx
kx e
e dx C
k
 xx
e dx e C

20. 20
Shortcuts: Integrals of
Expressions Involving ax + b  
 
 
1
1
( 1)
n
n ax b
ax b dx C n
an


    

  
1 1
lnax b dx ax b C
a

    1
ax b ax b
e dx e C
a

 1
ln
ax b ax b
c dx c C
ac



21. Examples 9 99
tt t
e dt e dteC 9
9
9

t
te
e dt C 5
5
4
4
33
5
4
u
u e
e du C





 55
44
4
3
5
12
5
u u
e eCC








22. You Do 5
2
x
dx

 5
2
5(ln2)
x
C




23. Indefinite Integral of x
-1

The rule is
23 1 1
lnx dx dx x C
x

  
Note: if x takes on a negative value, then
ln x will be undefined. The absolute value
sign keeps that from happening.

24. Example 4l
1
n
4
4dx dx
x
x
x
C 
You Do: 25
x
e dx
x




 21
5ln
2
x
x e C

  
Integrating Exponential Functions.mp4

25. 25
Exercises
1. ??????�
−??????
2
�??????
2. 3�
4??????
�??????
3. �
??????
4
4??????
3
�??????
4.
4�??????
sec??????�
????????????????????????

5. 7
2??????+3


Evaluate the following.

26. 1. �
1
2
??????
�??????
2. 6�
−3??????
�??????
3. �
3??????+1
�??????
4. 5??????�
−??????
2
�??????
5.
�
1
??????
??????
2
�??????

26
assignment
Evaluate the following functions.

27. 27
1.

2.
3.

4.

5. 3
3
2
t
t
e dt
e
Evaluate the following functions.
ADDITIONAL Exercises

28. 6.

7.

8.

9.
??????+1
??????
�??????

10.
28 dx
x
x







2
1 3
(ln )x
dx
x


29. Thank You!
29