Integral

1. Quote of the Day
“The only way to learn
mathematics is to do
mathematics.” – Paul Halmos

2. CALCULUS 2
EE 1-A and EE 1-B
Instructor:
ENGR. BENJIE P. ICAIN, MAT Mathematics

3. LESSON 2 – Indefinite Integral
In this lesson, you should be able to:
 solve antiderivatives using the properties of
integration and by substitution
 solve applied problems using the properties of
integration and by substitution method

4. Indefinite Integral
𝑰𝒇 𝑭′
𝒙 = 𝒇 𝒙 , 𝑡ℎ𝑒𝑛 𝒇 𝒙 𝒅𝒙 = 𝑭 𝒙 + 𝑪
For any real number C.
Note:
The symbol 𝑖𝑠 the integral sign, 𝑓(𝑥) is the integrand, and
𝑓(𝑥)𝑑𝑥 is called an indefinite integral, the most general
antiderivative of 𝑓.

5. Power Rule
𝒙𝒏 𝒅𝒙 =
𝒙𝒏+𝟏
𝒏 + 𝟏
+ 𝑪, 𝒏 ≠ −𝟏

6. Constant Multiple Rule and Sum and Difference Rule
𝒄 ∙ 𝒇 𝒙 𝒅𝒙 = 𝒄 𝒇 𝒙 𝒅𝒙 , for any real number c
(a constant maybe written before the integral sign)
and
𝒇 𝒙 ± 𝒈 𝒙 𝒅𝒙 = 𝒇 𝒙 𝒅𝒙 ± 𝒈 𝒙 𝒅𝒙
(The integral of the sum/difference of two functions is the
sum/difference of their integrals.)

7. EXAMPLEs
Find the indefinite integral of the following
1. 𝑢5
𝑑𝑢
SOLUTION Use the power rule with n = 5
𝑢5𝑑𝑢 =
𝑢5+1
5 + 1
+ 𝐶 =
𝒖6
6
+ 𝑪
To check the solution, find the derivative of
𝑢6
6
+ 𝐶. The derivative is
𝑢5, the original function
𝒙𝒏 𝒅𝒙 =
𝒙𝒏+𝟏
𝒏 + 𝟏

8. 𝟐.
𝟏
𝒙𝟑
𝒅𝒙
SOLUTION First, write
1
𝑥3 𝑎𝑠 𝑥−3. Then, n = - 3
1
𝑥3
𝑑𝑥 = 𝑥−3
𝑑𝑥 =
𝑥−3+1
−3 + 1
+ 𝐶 =
𝑥−2
−2
+ 𝐶
= −
1
2𝒙2
+ 𝑪
Verify the solution by differentiating −
1
2𝑥2 + 𝐶 to get
1
𝑥3

9. 𝟑. 𝟑 𝒙𝟑 𝒅𝒙
SOLUTION Since 𝑥3 = 𝑥
3
2, then 𝑛 =
3
2
3 𝑥3 = 3 𝑥
3
2 𝑑𝑥 = 3
𝑥
5
2
5
2
+ 𝐶
=
6
5
𝒙
5
2 + 𝑪 =
6
5
𝒙𝟓
+ 𝑪

10. 4. 5𝑤4
−3𝑤2
+ 2 𝑑𝑤
SOLUTION Integrate each term separately by extending the sum
and difference rule to more than two terms, we get
5𝑤4
−3𝑤2
+ 2 𝑑𝑤 = 5 𝑤4
𝑑𝑤 − 3 𝑤2
𝑑𝑤 + 2 𝑑𝑤
= 5
𝑤4+1
4 + 1
− 3
𝑤2+1
2 + 1
+ 2 𝑤 + 𝐶
= 5
𝑤5
5
− 3
𝑤3
3
+ 2𝑤 + 𝐶
= 𝒘5
− 𝒘3
+ 2𝒘 + 𝑪

11. 5.
𝑥3
− 2
𝑥
𝑑𝑥
SOLUTION Rewrite the integrand, we get
𝑥3
− 2
𝑥
𝑑𝑥 =
𝑥3
𝑥
−
2
𝑥
𝑑𝑥 Rewrite as difference of fraction
=
𝑥3
𝑥
1
2
−
2
𝑥
1
2
𝑑𝑥 𝑥 = 𝑥
1
2
= 𝑥
5
2 − 2𝑥
−1
2 𝑑𝑥 use
𝑥𝑚
𝑥𝑛 = 𝑥𝑚−𝑛
=
2
7
𝒙
7
2 − 4𝒙
1
2 + 𝑪

12. 6. 𝑡2
+ 5 2
𝑑𝑡
SOLUTION Expand the binomial first, and find the indefinite integral.
𝑡2
+ 5 2
𝑑𝑡 = 𝑡4
+ 10𝑡2
+ 25 𝑑𝑡
=
𝑡5
5
+
10
3
𝑡3
+ 25𝑡 + 𝐶

13. 7. Suppose a publishing company has found that the
marginal cost (cost added by producing one additional
unit of a product or service) at a level of production of x
thousand books is given by
𝐶′
𝑥 =
50
𝑥
And that the fixed cost (the cost before the first book can
be produced) is ₱25,000. Find the cost function 𝐶(𝑥) and
the cost of producing 400 books.

14. SOLUTION Use the indefinite integral rules to integrate the function
𝐶 𝑥 =
50
𝑥
𝑑𝑥 = 50𝑥
−1
2 𝑑𝑥 = 50
𝑥
1
2
1
2
+ 𝐾 = 100𝑥
1
2 + 𝐾
𝐶 𝑥 = 100𝑥
1
2 + 𝐾
The constant of integration is K is
use to avoid confusion with the cost
function 𝐶(𝑥). To find the value of
K, use the fact that 𝐶(0) is ₱25,000
𝐶 𝑥 = 100𝑥
1
2 + 𝐾
25,000 = 100 ∙ 0 + 𝐾
𝐾 = 25,000
Thus, the cost function is 𝑪 𝒙 = 𝟏𝟎𝟎𝒙
𝟏
𝟐 + 𝟐𝟓, 𝟎𝟎𝟎

15. Thus, the cost function is 𝑪 𝒙 = 𝟏𝟎𝟎𝒙
𝟏
𝟐 + 𝟐𝟓, 𝟎𝟎𝟎
when 𝑥 = 400, in solving for the cost of producing books 𝐶(𝑥)
𝐶 400 = 100 400
1
2 + 25,000
𝐶 400 = 100 400
1
2 + 25,000
𝐶 400 = 2000 + 25,000
𝐶 400 = 27,000
The cost in producing 400 books is ₱ 27,000

16. 24𝑥7
−15𝑥2
+
4
𝑥3
𝑑𝑥
= 3 𝑥8
− 5𝑥3
−
2
𝑥2
+ 𝐶

17. 1. Find the cost function for each marginal
cost function.
𝑎. ) 𝐶′ 𝑥 = 3𝑥 − 1, fixed cost is ₱40
𝐶 𝑥 =
3𝑥2
2
− 𝑥 + 40

18. To GOD be the GLORY!!!
THANK YOU!