· To develop an understanding of indefinite and definite integration by sight and by u substitution. Supported by learning objectives 1 and 2.
| H.1 |
| H.2 |
| H.3 |
Objective
H.1 Back to Top
To integrate functions by sight.
Instructional Notes
To this point in the course, students have learned to differentiate functions and apply their knowledge of derivatives to graphing functions and solve a wide variety of problems. By reversing the process of differentiation, that is beginning with the derivative of a function and determining the function, many new kinds of problems can be solved. This reverse process is known as integration or antidifferentiation. Discuss the meaning of the word “integration”.
In this objective, students are to determine a function, from its derivative, by simply looking at the derivative and using “intelligent guessing”. Begin with a few examples such as the following:
Example: Find a function 
whose derivative, 
, is 
.
Solution: Given time, the students should respond with 
. They will not likely come up with the 
immediately, but if you tell them that the answer 
is not quite complete, you will likely have someone say that the answer could be 
plus some number. Point out to the students that the 
does not necessarily mean that the constant is positive.
As you develop the examples, ask students to observe the powers of x and the coefficients of x in 
compared to the powers and coefficients of x in 
through this and the subsequent examples.
Example: Find a function 
such that 
.
Solution: 
.
Example: Find a function y whose differential, 
, is 
.
Solution: The students will need to recall the definition of the differential, namely that 
. In this case, 
. Thus 
. It follows, then, that 
.
Example: Find the function y for which 
.
Solution: Since 
, 
. Students need to recall that if 
, 
. They could then begin by hypothesizing that 
. They must check by differentiating. If 
, 
. The result is 3 times larger than it should be so the hypothesis needs modification. Hopefully, they will next try 
whose differential is indeed 
.
In the early examples, avoid situations in which the function to be integrated involves 
. Examine this more closely after the introduction of the integral sign.
Introduce students to the integral sign, namely 
. Point out that to indicate the operation of integration or antidifferentiation, the symbol 
is placed in front of the expression. The symbol 
can mean either of the following:
• What function has 
as its derivative if differentiated with respect to x ?
• What function has 
as its differential?
Thus the question “find a function 
whose derivative, 
, is 
” can be abbreviated by just writing 
.
Point out that the integral sign is an elongated S and comes from the fact that integration and “Sums (Sigma)” are very much related although students have not seen that relationship yet.
It is now time to deal with how to find 
if 
, or using our integral notation, what is 
? If students think that to integrate power functions, one raises the power of x by 1 and then divides by the new power, you can expect students to respond with 
. Some will immediately recognize that this result makes no sense since division by 0 is undefined. Given a bit more time, someone may respond with 
. This is the best response that a student could give at this time, but not quite complete. Students cannot be faulted for the incomplete response because, unless the teacher tackled the suggestions/extensions column in G.1, students would not realize that 
also has derivative 
. It is time to bring this result out into the open.
Theorem: If 
, then 
.
Proof:
If 
, then 
, and the result follows since 
.
If 
, then 
, and 
. Then,
A more complete response to 
is 
. This allows for situations in which x may be negative.
Example: Find 
.
Solution: Rewrite the integral as 
. Then the result follows: 
.
Stress the importance of checking the answer by differentiating .
Since we have established several rules for differentiation, we can also establish the equivalent rules for integration. This is a good time for the class to put the rules side by side, discuss each, and create a formula sheet.
If 
, 
, 
If 
, 
If 
, 
If 
, 
*
If 
, 
*
If 
, 
*
If 
, 
*
If 
, 
If 
, 
If 
, 
If 
, 
#
If 
, 
#
If 
, 
#
* These are optional
#Remember that differentiating inverse trigonometric functions is an optional part of the curriculum.
Usually the formula 
is omitted from a table of integrals because 
has the same derivative as 
. This does not mean that 
. It just means that they have the same tangent line slopes at the same u values.
In an informal way (through the use of examples), establish the following general integration principles.
Integration Principles 
Vocabulary
is read as “the integral of 
with respect to x ”. The 
serves to identify x as the variable of integration.
If 
, then C is the constant of in t egration, 
is the integrand and 
is the antiderivative or indefinite integral .
As additional instructional examples are developed, emphasize the importance of writing the expressions to be integrated in the form 
, or sums and differences thereof, whenever possible. This is the same form in which we tried to write functions to perform differentiation before we knew about the product, quotient, and chain rule.
Example: 
Solution: 
Example: 
Solution:
Examples/Activities
Find a function 
having the given derivative.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
You have shown students that if 
, then 
. Students should practise using this result, since it is new to them.
Find 
.
1. 
Solution: 
2. 
3. 
4. 
5. 
Tackle several questions of the following type. It is important to prepare students to integrate expressions of the form 
by sight.
1. 
2. 
3. 
5. 
6. 
7. 
where a and b are constants
Evaluate each of the following integrals by sight. Mentally check each answer by differentiating. Omit 16 to 21 if you did not work with all of the trigonometric functions.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
Perform each of the following integrations.
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
(Long division yields 
. Thus 
.)
17. 
18. 
(Hint: 
)
19. 
(Hint: Recall that 
, thus 
.)
20. 
(Hint: 
)
Do 21 and 22 only if you have differentiated all the trigonometric
functions.
21. 
(Hint: 
)
22. 
Hint: 
Suggestions/Extensions
Have the students discuss how differentiation and integration are inverses of each other and have students compare how these inverses are related to other inverses they have studied.
Have the students work in pairs and discuss how they are thinking through the integrations (COM, PSVS).
Many students confuse derivatives and integrals. By working in pairs and discussing their thinking, the students can clarify their own thinking, and verify the thinking and logic for each other.
Students could differentiate 
and 
. Based on these results, students can complete the following integration formulas:
The following type of problem is a reasonable challenge for students. The problem represents an introduction to that part of calculus known as solving differential equations.
Find the function 
for which 
, 
, and 
.
Solution:
If 
, then 
. Since 
, we have 
. Solving, 
. Thus, 
. Then, 
. Since 
, we have 
. Solving, 
. Thus, 
.
Objective
H.2 Back to Top
To integrate functions using u substitution.
Instructional Notes
As a motivational set for this objective, ask students to perform the following integral by sight: 
. Follow up each of their attempts by differentiation to see if they are correct. Most students will realize that determining this integral by sight is beyond their capabilities. Tell students that the question given fits one of the integration formulas given in H.1 and provide more time for students to find the integral. It is unlikely that they will find the integral. If they do not, ask them to differentiate 
. They should obtain 
. Again, ask them to which of the integral formulas this question seems to be similar. Students may still be unsure. Show them how this question is modelled after the formula 
, where 
. Moreover, if 
, then the differential of u , namely 
, which by definition is 
, becomes 
. Thus, the original question 
is really like 
or 
. The main purpose of the motivational set is to show students that integration cannot always be performed mentally. There are integration strategies that can be learned to make integration simpler, just as there were differentiation strategies (product rule, quotient rule, chain rule, etc.) that simplified differentiation. Although students can differentiate almost any function that can be written, it is a completely different story when it comes to integration. For example, using a product rule we can easily differentiate 
, but try performing the integral 
. (This integral actually can be performed and after considerable work yields: 
). There are many integrals, however, that are extremely difficult to perform. Try the innocent looking question 
.
As you develop the technique of integration using u substitution, students should have an integration formula sheet in front of them. Students should view the formulas as moulds or models. The challenge for students is to convert the given integral into one of these moulds by using an appropriate substitution. Students need to use the letter u to replace part (or all) of the integrand. Then, they must find the expression for the differential 
and use this expression (or a numerical multiple of this expression) to replace any portion of the integrand that u has not replaced. The difficulty is trying to determine what u should represent. Initially, trial and error is part of this technique. After students gain experience, they will make more intelligent guesses for u . In the examples that follow, there are some pointers that can be shared with the students. An obvious pointer, even before starting any examples, is that there is no value in letting u represent x . This does nothing to change the form of the original questions.
Begin with an example that is simple enough for students to do by sight. Example 1: Find 
.
Solution:
Ask students to perform the integration mentally. Someone will likely suggest 
. Check by differentiating. It is out slightly by a factor of 3. How must the estimate be modified to be correct? Should the estimate be 
as large or 3 times bigger? After some more experimentation, students will arrive at the correct result 
.
Another method to perform this integration, although complex is to expand 
and then integrate each of the resulting 30 terms.
Now that students know the answer, have them try the technique of u substitution.
Have students consider how this integral has a similar format to 
. This format occurs frequently. Since our question contains a variable base raised to a numerical power, this representative integral seems appropriate. Let 
.
Then 
Find 
. Remember that if 
, then 
. Here, 
. Thus, 
. Another way to obtain the result is if 
, then 
, so 
.
Moreover, 
.
Substitute into the integral for x and dx using the expressions for u and du :
.
Remove the constant using the constant multiple rule of integration: 
.
.
Integrate with respect to u . Note that you do not have to multiply C by 
since 
is a constant:
.
Simplify, and replace u by 
. The result agrees with our earlier “by sight” answer:
.
Example 2: Find 
.
Solution:
.
Have the students suggest how they might deal with the format of the integral and compare it to the formats seen before. Ultimately, have the students recognize how the integral can be written as 
which looks like 
.
Assign u and immediately find du . u has been correctly assigned if du is a numerical multiple of that portion of the integrand that was not assigned to u . Let 
. Then 
. Here 
which is a multiple of 
, that portion of the integrand that was not assigned to u . Adjust the expression for du to get 10 dx in the original question.
Substitute, replacing 
by 
and 
by 
. 
Remove the constant multiple.
Perform the integration.
Express the answer in terms of x by substituting for u.
Example 3: Find 
.
Solution:
What should u be this time? Is there a power of an expression in this example as there was in the previous two examples?
Try letting 
. Then 
.
Substituting in, we obtain 
. Notice that du is exactly equal to everything in the integrand that was not assigned to u .
Perform the integration.
Express the answer in terms of x by substituting for u .
Example 4: Find 
.
Solution: 
.
Note that the question contains e raised to a variable power. This looks like 
.
Let 
.
Find 
: 
.
Note that du is a numerical multiple of that part of the integrand that was not assigned to u , that is 
is a multiple of 
.
Make the substitutions: 
.
Remove the constant multiple: 
.
Integrate: 
.
Write the answer in terms of x : 
.
Remind the students that as they rearrange the integrand, they should never have u 's and/or du 's mixed with x 's and/or dx 's. In terms of example 4, do not write 
followed by 
.
Example 5: Find 
.
Solution: 
.
There are many expressions we might try for u . The question does contain the sine of an expression. Let's see if we can use the mould 
.
Let 
.
Find du : 
.
Note that du is a multiple of that part of the integrand that was not assigned to u .
Adjust the du expression: 
.
Substitute: 
.
Remove the constant multiple, integrate, and write your answer in terms of x :
.
Examples/Activities
For the first few questions, tell the students what u should represent.
Integrate using u substitution.
1. 
, let 
2. 
, let 
3. 
, let 
4. 
, let 
5. 
, let 
6. 
, let 
7. 
, let 
8. 
, let 
9. 
, let 
10. 
, let 
Integrate each of the following using u substitution (16-20 require a knowledge of all the trigonometric function derivatives).
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
Integrate each of the following using u substitution (continued). Note: Number 33 requires a knowledge of the derivatives of trigonometric functions other than sine and cosine.
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
Suggestions/Extensions
Other more complicated u substitutions could be attempted. Consider the following:
Let 
. Then 
. Also since 
, 
. The question becomes 
. Rewriting, we have 
. The integral can now be found easily.
For additional challenge, try 
. Let 
, then 
or 
. Also 
. Think of the question as 
and make the following substitutions.
. This leads to 
and the eventual solution.
Allow students time to try letting u represent different parts of the integrals in order to help the students develop their visual clues and strategies for how to approach new integrals. When told what to use as u , ask the students to explain why they made that choice (CCT).
Students needing a greater challenge could be introduced to the technique of trigonometric substitution .
Consider 
.
If we let 
, that is 
, then 
. The question becomes:
We can visualize the substitution using a right triangle with hypotenuse 2 and one leg of length x . Then the length of the other leg will be 
.
Thus 
and 
. The final form of the answer becomes 
Objective
H.3 Back to Top
To perform definite integration.
To use language as a tool for learning and communicating (COM).
To develop and demonstrate the abilities to communicate in ways that support social harmony (PSVS).
Instructional Notes
In order to make the transition into the subsequent unit on applications of integration to finding areas, show the students how to perform definite integration. As you carry out some of the definite integrations, tell students how this will relate to the next unit.
What is definite integration?
If 
is a continuous function defined on 
, and if 
, then 
is called the definite integral of 
from a to b . We use the notation 
. Read this as “the integral from a to b of 
with respect to x is equal to 
”. The numbers a and b are called the upper and lower bounds of integration respectively. Often we write 
in place of 
.
In this unit, we will always arrive at a numerical value for the answer – hence, the name definite integral. The integration we performed in sections H.1 and H.2 was known as indefinite integration . There is an analogy to differentiation here. We can find the slope of the tangent line to a curve at a particular point and obtain a specific value, a number, or we can just obtain an expression for the slope of the tangent line at any point along the curve.
An example.
Indefinite Integration
Definite Integration
|
This is how the question appears. Note that
Integrate, leaving out the
Replace x by the upper bound of integration and subtract from this the value obtained by replacing x by the lower bound of integration.
Simplify.
|
|
There is no point in recording the 
because if we had, we would obtain 
, the same result as before because 
.
Point out to the students that in section I.1 we will prove that the value of the definite integral is actually the area bounded by the curve 
, the x- axis, and the vertical lines 
and 
as shown below.
Example: Find 
.
Solution: Note that 
is continuous on 
, so definite integration can proceed. The integral is a bit more challenging so we will use u substitution.
Let 
. Then 
and thus 
. Note that the bounds of integration, from 0 to 1, apply to the variable x . Since we are now working in the variable u , we change the bounds of integration accordingly. If 
, 
. If 
, 
. Thus, 
.
Example: 
Solution: We may not write that 
because 
is not continuous on 
, since it is undefined at 
. Thus, the definite integration cannot be performed with meaning.
Examples/Activities
Perform each of the following definite integrations. In pairs, have students take turns explaining the steps of their solutions (e.g., one student explains the solutions for the even-numbered questions while the other student explains the solutions for the odd-numbered questions). Prior to working in pairs, have students brainstorm (as a class) aspects of respectful communication that will support their work in pairs (e.g., understand and avoid roadblocks to communicating clearly and constructively such as assuming to know rather than asking, confronting or cornering rather than seeking to find common ground) (PSVS).
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. It can be shown that if 
is the velocity of a car in km/h at time t , then the distance the car has travelled from time 
hours to time 
hours is given by the definite integral 
. Find the distance travelled by a car from time 
hours to time 
hours for the following velocity functions.
(a) 
(b) 
(c) 
(d) 
• It can be shown that the average value of a continuous function 
over the closed interval 
is given by 
.
• Find the average value of the function 
from 
to 
.
• Suppose that the temperature in Moose Jaw on a July day is given by the function 
, where x is the number of hours that have passed by since 10:00 A.M. Find the average temperature in Moose Jaw between 11:00 A.M. and 3:00 P.M.
• If the sales of a business are growing continuously according to the function 
where 
is the sales in dollars on the t th then the total of all sales between the a th and the b th day is given by the 
. Find the total of all sales between 
and 
.
Suggestions/Extensions
Integrals such as 
are called improper integrals . If 
is continuous on the interval 
, the integral 
is defined as follows: 
. Show that 
, while 
does not exist.
is continuous on 
so definite integration can proceed. Always check for continuity. 
. Follow the integral by drawing a vertical line. Transfer the bounds of integration from the integral sign to the vertical line.