Theorem 4 of basic fundamental of integral
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This document summarizes the key theorems of integral calculus taught in a lesson. It outlines 4 theorems: 1) The integral of any variable dx is x + C 2) The integral of a constant times a variable is that constant times the variable + C 3) The integral of a variable raised to a constant power n is that variable raised to n+1 over n+1 + C, when n does not equal -1 4) The integral of 1 over any variable or a variable raised to a negative constant power is the natural log of the absolute value of that variable + C It provides examples for each theorem and has sections on reviewing the lesson, introducing a concept with a "magic
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