Theorem 4 of BASIC FUNDAMENTAL OF INTEGRAL

1. Theorem 4 of
BASIC
FUNDAMENTAL
OF INTEGRAL
MAGCALAS, Rommel
22 May 2017

2. Today’s Session
▪ Review of the Past Lesson
▪ Magic Box (Motivation)
▪ Lesson for the day
▪ Application
▪ Evaluation
▪ Objective for today’s session
Magcalas, Rommel DemoTeaching

3. Objective for today’s session
Evaluate the anti-
derivatives of a function /
integral of a function using
substitution rule and table of
integrals
Magcalas, Rommel DemoTeaching

4. Review of the Past Lesson
▪ What is INTEGRAL CALCULUS
▪ BASIC FUNDAMENTAL OF INTEGRAL
(Theorem 1-3)
Magcalas, Rommel DemoTeaching

5. INTEGRAL CALCULUS
▪ Branch of calculus that deals
with the function to be
integrated.
∫ f(x) dx
∫ = integral sign/symbol
f(x) = integrand
dx = variable of integration
Magcalas, Rommel DemoTeaching

6. BASIC FUNDAMENTAL
OF INTEGRAL
▪ THEOREM 1 (Integral of any
variable integration)
∫ dx = x + C
Magcalas, Rommel DemoTeaching

7. THEOREM 1 (Integral of any variable integration)
∫ dx = x + C
1. ∫ dy
2. ∫ dz
3. ∫ dw
4. ∫ dk
5. ∫ dp
= y + c
= z + c
= w + c
= k + c
= p + c
Magcalas, Rommel DemoTeaching

8. BASIC FUNDAMENTAL
OF INTEGRAL
▪ THEOREM 1 (Integral of any variable
integration)
∫ dx = x + C
▪ THEOREM 2 (Integral of any
constant number in any
variable integration)
∫ adx = ax + C
Magcalas, Rommel DemoTeaching

9. THEOREM 2 (Integral of any constant number in any
variable integration)
∫ adx = ax + C
1. ∫ 2dx
2. ∫ 5dg
3. ∫ 12dk
4. ∫ 9dz
5. ∫ 7dw
= 2x + c
= 5g + c
= 12k + c
= 9z + c
= 7w + c
Magcalas, Rommel DemoTeaching

10. BASIC FUNDAMENTAL
OF INTEGRAL
▪ THEOREM 1 (Integral of any variable
integration)
∫ dx = x + C
▪ THEOREM 2 (Integral of any constant number in
any variable integration)
∫ adx = ax + C
▪ THEOREM 3 (Integral of any variable raise to a
constant number in a variable integration)
∫ xndx = xn+1 + C
n + 1
Note: n ≠ -1Magcalas, Rommel DemoTeaching

11. THEOREM 3 (Integral of any variable raise to a
constant number in a variable integration)
∫ xndx = xn+1 + C
n + 1
1. ∫ x3dx
2. ∫ w5dw
3. ∫ z9dz
4. ∫ k19dk
5. ∫ y32dy
= x3+1 + C = x4 + C
3 + 1 4
= w5+1 + C = w6 + C
5 + 1 6
= z9+1 + C = z10 + C
9 + 1 10
= k19+1 + C = k20 + C
19 + 1 20
= y32+1 + C = y33 + C
32 + 1 33
Magcalas, Rommel DemoTeaching

12. BASIC FUNDAMENTAL
OF INTEGRAL
▪ THEOREM 1 (Integral of any variable
integration)
∫ dx = x + C
▪ THEOREM 2 (Integral of any constant number in
any variable integration)
∫ adx = ax + C
▪ THEOREM 3 (Integral of any variable raise to a
constant number in a variable integration)
∫ xndx = xn+1 + C
n + 1
Note: n ≠ -1Magcalas, Rommel DemoTeaching

13. Magic Box
Magcalas, Rommel DemoTeaching

14. Magic Box
1. Pass the box to your seatmate
while the music is continuously
playing
2. Once the music stops, picked a
card inside it
3. Categorize the examples into
what Theorem it belongs
INSTRUCTION for this Game
Magcalas, Rommel DemoTeaching

15. Magic Box
Magcalas, Rommel DemoTeaching

16. BASIC FUNDAMENTAL
OF INTEGRAL
▪ THEOREM 1
∫ dx = x + C
▪ THEOREM 2
∫ adx = ax + C
▪ THEOREM 3
∫ xndx = xn+1 + C
n + 1
Note: n ≠ -1
THEOREM 4 (Integral of
1 over any variable in
a variable integration
or Integral of any
variable raise to
negative 1 in a
variable integration)
∫ 1 dx
x
= ∫ x-1 dxMagcalas, Rommel DemoTeaching

17. = ∫ x-1 dx
= ln │x│ + C
ABSOLUTE VALUE | |
▪ to remove any sign in front of a
number and think of all numbers as
positive (or zero).
∫ 1 dx
x
Magcalas, Rommel DemoTeaching

18. BASIC FUNDAMENTAL
OF INTEGRAL
▪ THEOREM 1
∫ dx = x + C
▪ THEOREM 2
∫ adx = ax + C
▪ THEOREM 3
∫ xndx = xn+1 + C
n + 1
Note: n ≠ -1
THEOREM 4 (Integral of 1
over any variable in a
variable integration or
Integral of any variable
raise to negative 1 in a
variable integration)
∫ 1 dx
x
= ∫ x-1 dx
= ln │x│ + CMagcalas, Rommel DemoTeaching

19. THEOREM 4 (Integral of 1 over any variable in a variable
integration or Integral of any variable raise to any negative
constant number in a variable integration)
∫ 1 dx = ln │x│ + C
x
1. ∫ 1 dx
q
2. ∫ 3 dy
y
3. ∫ 11 dw
w
4. ∫ 7 dx
x
5. ∫ 9 dz
z
= ln | q | + C
= ∫ 3 . 1 dy = 3 ln | y | + C
y
= ∫ 11 . 1 dw = 11 ln | w | + C
w
= ∫ 7 . 1 dx = 7 ln | x | + C
x
= ∫ 9 . 1 dz = 9 ln | z | + C
z
Magcalas, Rommel DemoTeaching

20. THEOREM 4 (Integral of 1 over any variable in a variable
integration or Integral of any variable raise to any negative
constant number in a variable integration)
∫ 1 dx = ln │x│ + C
x
1. ∫ 4 dx
3x
2. ∫ x2-2x +1 dx
x2
3. ∫ 1 dx
x + 2
4. ∫ 3x dx
x2
5. ∫ 8 + 2y – 10y-2 dy
y
= ∫ 4 . 1 dx = 4 ln | x | + C
3 x 3
= ∫x2 dx - ∫2x dx + ∫1 dx
x2 x2 x2
= x- ∫2 . 1 dx + x-2+1 dx
x -2+1
= x - 2 ln |x| - 1 + c
x
= ln │x + 2│+ C
= 3 ln │x│+ C
= 8 ln │y│+ 2y + 20 + C
y2Magcalas, Rommel DemoTeaching

21. APPLICATION:
Evaluate the following integrals:
1. ∫ 3x-1 dx
2. ∫ 5 -4y + 6y-2 dy
2y
3. ∫ 2y dy + 2 dy
y
Magcalas, Rommel DemoTeaching

22. EVALUATION:
Evaluate the following integral:
1. ∫ 3x-1 dx
2. ∫ 1 dx
3x
3. ∫ 2x-1 + 2 dx
4. ∫ 6z - 4z2 + z-1 dz
z3
Magcalas, Rommel DemoTeaching

23. ASSIGNMENT:
Evaluate:
1. ∫ 1 + x + x-1+x-2+x-3 dx
Magcalas, Rommel DemoTeaching

24. That’s all for today!
THANK YOU!!!
Magcalas, Rommel DemoTeaching