Integral

1. Review:
Find the derivative of the following.
1. 𝑦 = 𝑥4
− 3𝑥
2. 𝑦 = 3𝑥3
+ 2𝑥2
− 10

2. Group Activity
1.
𝑑
𝑑𝑥
(𝑥3
+𝑥2
− 16)
𝑑
𝑑𝑥
=
𝑑
𝑑𝑥
𝑥3
+
𝑑
𝑑𝑥
𝑥2
−
𝑑
𝑑𝑥
16
𝑑
𝑑𝑥
= 3𝑥3−1
+ 2𝑥2−1
− 0
𝑑
𝑑𝑥
= 𝟑𝒙𝟐
+ 𝟐𝒙

3. 2. (3𝑥2
+ 2𝑥)𝑑𝑥
3𝑥2
𝑑𝑥 + 2𝑥𝑑𝑥
3 𝑥2
𝑑𝑥 + 2 𝑥𝑑𝑥
3 𝑥 2+1
2+1
+
2 𝑥1+1
1+1
+ 𝐶
3𝑥3
3
+
2𝑥2
2
+ 𝐶
= 𝒙𝟑
+𝒙𝟐
+ 𝑪

4. • What have you observed on the two
solutions?
• Identify the differences between the two
solutions, start with the exponent.
• What do you think is Antiderivatives all about?

5. Antiderivatives

6. OBJECTIVES
At the end of the lesson, the students will be
able to:
a. illustrate antiderivative of a function
b. determine the rules of antiderivatives
c. solve problems involving antiderivatives

7. Antiderivatives
- a function such that its derivative
equals the original function

8. Power rule
𝑥𝑛
𝑑𝑥 =
𝑥𝑛+1
𝑛 + 1
+ 𝐶

9. Example 1: 𝒙𝟐𝒅𝒙
𝑥2𝑑𝑥 =
𝑥2+1
2+1
+ 𝐶
=
𝑥3
3
+ 𝐶
=
𝟏
𝟑
𝒙𝟑 + 𝑪

10. Example 2: 𝒙𝟒
𝒅𝒙
𝑥4𝑑𝑥 =
𝑥4+1
4+1
+ 𝐶
=
𝑥5
5
+ 𝐶
=
𝟏
𝟓
𝒙𝟓 + 𝑪

11. Example 3: 𝒙
𝟏
𝟒𝒅𝒙
𝑥
1
4𝑑𝑥 =
𝑥
1
4
+1
1
4
+1
+ 𝐶
=
𝑥
5
4
5
4
+ 𝐶
=
𝟒
𝟓
𝒙
𝟓
𝟒 + 𝑪

12. Constant Multiple Rule
𝑎 ∙ 𝑓 𝑥 𝑑𝑥 = 𝑎 𝑓 𝑥 𝑑𝑥

13. Example 1: 𝟔𝒙𝟐𝒅𝒙
6𝑥2
𝑑𝑥 = 6 𝑥2
𝑑𝑥
= 6 ∙
𝑥2+1
2+1
+ 𝐶
= 6 ∙
𝑥3
3
+ 𝐶
= 𝟐𝒙𝟑
+ 𝑪

14. Example 2: 8𝑥
3
2𝑑𝑥
8𝑥
3
2𝑑𝑥 = 8 𝑥
3
2𝑑𝑥
= 8 ∙
𝑥
3
2+1
3
2
+1
+ 𝐶 = 8 ∙
2
5
𝑥
5
2 + 𝐶
= 8 ∙
𝑥
5
2
5
2
+ 𝐶 =
𝟏𝟔
𝟓
𝒙
𝟓
𝟐 + 𝑪

15. Sum and Difference Rule
𝒇 𝒙 ± 𝒈 𝒙 𝒅𝒙 = 𝒇 𝒙 𝒅𝒙 ± 𝒈 𝒙 𝒅𝒙

16. Example: 𝟑𝒙 + 𝟓 𝒅𝒙
3𝑥 + 5 𝑑𝑥
= 3𝑥𝑑𝑥 + 5𝑑𝑥
= 3 𝑥𝑑𝑥 + 5 𝑑𝑥
= 3 ∙
𝑥1+1
1+1
+ 5𝑥 + 𝑐
= 3 ∙
𝑥2
2
+ 5𝑥 + 𝑐
=
𝟑
𝟐
𝒙𝟐
+ 𝟓𝒙 + 𝒄

17. Example: 𝟐𝒙𝟑
+ 𝟒𝒙𝟐
+ 𝟏𝟎 𝒅𝒙
2𝑥3
+ 4𝑥2
+ 10 𝑑𝑥 = 2𝑥3
𝑑𝑥 + 4𝑥2
𝑑𝑥 + 10𝑑𝑥
= 2 𝑥3
𝑑𝑥 + 4 𝑥2
𝑑𝑥 + 10 𝑑𝑥
= 2 ∙
𝑥3+1
3+1
+ 4 ∙
𝑥2+1
2+1
+ 10𝑥 + 𝑐
= 2 ∙
𝑥4
4
+ 4 ∙
𝑥3
3
+ 10𝑥 + 𝑐
=
𝟏
𝟐
𝒙𝟒
+
𝟒
𝟑
𝒙𝟑
+ 𝟏𝟎𝒙 + 𝒄

18. Up and Down
1. 𝑥6
𝑑𝑥
2. 5𝑥3
𝑑𝑥
3. 5𝑥3
+ 5 𝑑𝑥
4. 𝑥𝑑𝑥
5.
𝑑𝑥
𝑥3
1.
𝑥7
7
+ 𝐶
2.
5
4
𝑥4
+ 𝐶
3.
5
4
𝑥4
+ 5𝑥 + 𝐶
4.
2
3
𝑥
3
2 + C
5. −
1
2𝑥2 + 𝑥

19. Quiz
Solve each item using Antidifferentiation.
1.
5𝑑𝑥
3
𝑥2
2. (3𝑥3
+ 2𝑥2
− 𝑥 + 5)𝑑𝑥
3. 𝑥3
(𝑥2
− 2)𝑑𝑥

20. Assignment
In your notebook, solve
𝑥−1 2
2 𝑥
𝑑𝑥.