This document provides formulas and examples for evaluating indefinite integrals involving various types of functions. It covers: 1) The integral of a variable to a power; 2) Integrals involving constants; 3) Integrals of sums and algebraic expressions; 4) Integrals of functions raised to a power; 5) Integrals of quotients using three different methods. It also provides shortcuts for integrals involving expressions of ax + b, integrals of exponential functions, and the integral of x-1. Examples are included to demonstrate each formula and concept.

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 b e 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) = ex then f’(x) = ex
 If f(x) = ax then f’(x) = (ln a)ax
 If f(x) = ekx then f’(x) = kekx
 If f(x) = akx then f’(x) = k(ln a)akx
This leads to the following formulas:

19. Indefinite Integrals of
Exponential Functions
19 ln
k x
kx a
a dx C
k a

 
ln
x
x a
a dx C
a
 
kx
kx e
e dx C
k
 
x x
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
a n


    

 
1 1
lnax b dx ax b C
a

   
1ax b ax b
e dx e C
a
 
 
1
ln
ax b ax b
c dx c C
a c
 
 

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

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. 𝑥𝑒−𝑥2
𝑑𝑥
2. 3𝑒4𝑥
𝑑𝑥
3. 𝑒 𝑥4
4𝑥3
𝑑𝑥
4.
4𝑑𝑥
sec 𝑥𝑒 𝑠𝑖𝑛𝑥
5. 72𝑥+3
Evaluate the following.

26. 1. 𝑒
1
2
𝑥
𝑑𝑥
2. 6𝑒−3𝑥
𝑑𝑥
3. 𝑒3𝑥+1
𝑑𝑥
4. 5𝑥𝑒−𝑥2
𝑑𝑥
5.
𝑒
1
𝑥
𝑥2 𝑑𝑥 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.
𝑥+1
𝑥
𝑑𝑥
10. 28
dx
x
x 






2
1
3
(ln )x
dx
x

29. Thank You!
29