2. Introduction Integration is the reverse process of Differentiation Differentiating gives us a formula for the gradient Integrating can get us the formula for the curve, if we know the gradient function It can also be used to calculate the Area under a curve

4. Integration You can integrate functions of the form f(x) = ax n where ‘n’ is real and ‘a’ is a constant Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment. If: 8A So integrating 2x should give us x 2, but we will be unsure as to whether a number has been added or taken away Differentiating Function Multiply by the power Reduce the power by 1 Gradient Function Integrating Function Divide by the power Increase the power by 1 Gradient Function

5. Integration You can integrate functions of the form f(x) = ax n where ‘n’ is real and ‘a’ is a constant Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment. 8A Integrating Function Divide by the power Increase the power by 1 Gradient Function Mathematically speaking… We increased the power by 1, then divided by the (new) power If: Then:

6. Integration You can integrate functions of the form f(x) = ax n where ‘n’ is real and ‘a’ is a constant Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment. 8A Integrating Function Divide by the power Increase the power by 1 Gradient Function Example Questions If: Then: Integrate the following: a) Increase the power by one, and divide by the new power DO NOT FORGET TO ADD C! b) Increase the power by one, and divide by the new power DO NOT FORGET TO ADD C!

7. Integration You can integrate functions of the form f(x) = ax n where ‘n’ is real and ‘a’ is a constant Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment. 8A Integrating Function Divide by the power Increase the power by 1 Gradient Function Example Questions If: Then: Integrate the following: c) Increase the power by one, and divide by the new power DO NOT FORGET TO ADD C! d) Increase the power by one, and divide by the new power DO NOT FORGET TO ADD C!

9. Integration You can apply the idea of Integration separately to each term of dy / dx  In short, if you have multiple terms to integrate, do them all separately 8B Example Question Integrate the following: Integrate each part separately ‘Tidy up’ terms if possible

10. Integration You can apply the idea of Integration separately to each term of dy / dx  In short, if you have multiple terms to integrate, do them all separately 8B Example Question Integrate the following: Integrate each part separately Deal with the fractions Rewrite if necessary

12. Integration You need to be able to use the correct notation for Integration 8C This the the integral sign, meaning integrate This is the expression to be integrated (brackets are often used to separate it) The dx is telling you to integrate ‘with respect to x’ Example Question Find: Integrate each part separately Deal with the fractions

13. Integration You need to be able to use the correct notation for Integration 8C This the the integral sign, meaning integrate This is the expression to be integrated (brackets are often used to separate it) The dx is telling you to integrate the ‘x’ parts Example Question Find: Integrate each part separately Deal with the fractions

14. Integration You need to be able to use the correct notation for Integration 8C This the the integral sign, meaning integrate This is the expression to be integrated (brackets are often used to separate it) The dx is telling you to integrate the ‘x’ parts Example Question Find: Integrate each part separately Deal with the fractions p and q 2 should be treated as if they were just numbers!

15. Integration You need to be able to use the correct notation for Integration 8C This the the integral sign, meaning integrate This is the expression to be integrated (brackets are often used to separate it) The dx is telling you to integrate the ‘x’ parts Example Question Find: Integrate each part separately

18. Integration You can find the constant of integration, c, if you are given a point that the function passes through Up until now we have written ‘c’ when Integrating. The point of this was that if we differentiate a number on its own, it disappears. Consequently, when integrating, we cannot be sure whether a number was there originally, and what it was if there was one… Step 1: Integrate as before, putting in ‘c’ Step 2: Substitute the coordinate in, and work out what ‘c’ must be to make the equation balance.. 8E Example Question The curve X with equation y = f(x) passes through the point (2,15). Given that: Find the equation of X. Integrate Sub in (2,15) Work out each fraction Add the fractions together Work out c

19. Integration You can find the constant of integration, c, if you are given a point that the function passes through Up until now we have written ‘c’ when Integrating. The point of this was that if we differentiate a number on its own, it disappears. Consequently, when integrating, we cannot be sure whether a number was there originally, and what it was if there was one… Step 1: Integrate as before, putting in ‘c’ Step 2: Substitute the coordinate in, and work out what ‘c’ must be to make the equation balance.. 8E Example Question The curve X with equation y = f(x) passes through the point (4,5). Given that: Find the equation of X. Split into 2 parts Write in the form ax n Integrate

20. Integration You can find the constant of integration, c, if you are given a point that the function passes through Up until now we have written ‘c’ when Integrating. The point of this was that if we differentiate a number on its own, it disappears. Consequently, when integrating, we cannot be sure whether a number was there originally, and what it was if there was one… Step 1: Integrate as before, putting in ‘c’ Step 2: Substitute the coordinate in, and work out what ‘c’ must be to make the equation balance.. 8E Example Question The curve X with equation y = f(x) passes through the point (4,5). Given that: Find the equation of X. Rewrite for substitution y = 5, x = 4 Work out each part carefully

21. Summary We have learnt what Integration is We have seen it combined with rewriting for substitution We have learnt how to calculate the missing value ‘c’, and why it exists in the first place