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Similar to Lesson 1 integration as the inverse of differentiation
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Lesson 1 integration as the inverse of differentiation
- 1. An introduction to Integration Theinverse process of Differentiation IB SL/HL
- 2. The inverse ofdifferentiation f (x) = 4x3 + 6x2 Notation for integration. Integration is the opposite of differentiation. 4x3 + 6x2 ( )dx∫ Integrate with respect to x. 4x3 + 6x2 ( )dx∫ = 4x4 4 + 6x3 3 = x4 + 2x3 Add one to the power and divide by the new power. Now differentiate and we should return to f(x). Also if we take a function, differentiate it and then integrate it we will return to the original function. For each of the functions below differentiate and then integrate with respect to x. 1. f (x) = 3x2 − 8x + 5 2. g(x) =10x4 − 3x2 + 4x −10 3. h(x) = 4 x +12x3 − y What conclusions can you draw?
- 3. Integration This is calledindefinite integration. The formula for this is: xn dx = xn+1 n +1 + C, n ≠ −1∫
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- 5. Integration questions Questions Answers sinx( )dx∫ = −cosx + c 4x2 + 5x −1( )dx∫ = 4x3 3 + 5x2 2 − x + c cos x( )dx∫ = sin x + c ex ( )dx∫ = ex + c
- 6. Multiples 6x3 − 3x2 +12x +9( )dx∫ can be re-written as, now it becomes easier to integrate, 3 2x3 − x2 + 4x + 3( )dx =∫ 3 2x3 − x2 + 4x + 3( )dx∫ Questions 1. −3sin x( )dx∫ 3. 10x2 − 20x3 + 25x4 ( )dx∫ 3cosx + c 4ln x + c 10x3 3 − 5x4 + 5x5 + c
- 7. Finding the cvalue y = x3 + 2x2 − 4x + c (1,4) 4 = (1)3 + 2(1)2 − 4(1) + c c = 5 y = x3 + 2x2 − 4x + 5 Questions Find f(x) for each of the following.
- 8. Finding the cvalue y = x3 + 2x2 − 4x + c (1,4) 4 = (1)3 + 2(1)2 − 4(1) + c c = 5 y = x3 + 2x2 − 4x + 5 Questions Find f(x) for each of the following. f (x) = x2 2 + 3x −1 f (x) = 3cos x + 2 f (x) = 5ln x + x − 3