WJEC Core 2 Integration. Section 1: Introduction to integration

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1 WJEC Core Integration Section : Introuction to integration Notes an Eamples These notes contain subsections on: Reversing ifferentiation The rule for integrating n Fining the arbitrary constant Reversing ifferentiation Integration is the reverse of ifferentiation. If you are given an epression for, an you want to fin an epression for y, you nee to use integration. This is sometimes calle solving a ifferential equation. Because ifferentiating a constant term gives zero, when you integrate, you must always a an arbitrary constant. For eample, ifferentiating the functions y = ², y = ² +, y = ² etc all give y integrating gives ² + c, where c is a constant. Eample shows how you can integrate a function by thinking about what function you woul nee to ifferentiate to obtain the given function. Eample Fin y as a function of for each of the following. (i) y (iv) (v) (vi) y (i) y c The erivative of is. So integrating gives. Therefore integrating gives. y c The erivative of is 7. 7 So integrating 7 gives. 7 Therefore integrating gives 7. of /0/ MEI

2 WJEC C Integration Notes an Eamples y c The erivative of is. So integrating gives. Therefore integrating gives. (iv) (v) (vi) y c y c y c The erivative of is. So integrating gives. Therefore integrating gives, an integrating gives. The erivative of is. So integrating gives. Therefore integrating gives, an integrating gives. The erivative of is. So integrating gives. Therefore integrating gives. The rule for integrating n The metho for integrating any polynomial function can be summe up as: n n Integrating, where n is a positive integer, gives n n Integrating k, where n is a positive integer an k is a constant, n k gives n You can integrate the sum of any number of such functions by simply integrating one term at a time. This formula is true not only when n is a positive integer, but for all real values of n, incluing negative numbers an fractions, ecept for n = -. of /0/ MEI

3 WJEC C Integration Notes an Eamples The formula oes not work for n = -, since this woul give a enominator of 0. There is a ifferent way to integrate, which is covere in C. Eample Integrate each of the following graient functions. (i) ( )( ) (i) y c y c ( )( ) y c Remember the arbitrary constant As for ifferentiation, you also nee to be familiar with the use of notation f () when integrating, as in the net eample. Eample Fin f() in terms of for each of the following. (i) f ( ) f ( ) f ( ) (i) f ( ) f ( ) c n n by, i.e. multiply by. ivie of /0/ MEI

4 WJEC C Integration Notes an Eamples f ( ) f ( ) c f ( ) f ( ) c c n n, so ivie by. n n, so ivie by. n n ivie by, i.e. multiply by. You can see further eamples using the Flash resources Basic inefinite integration: single powers of, Integrating rational powers of an Basic inefinite integration: sums of powers of. Note that these resources use notation for integration which is not use in the tetbook until a little later: for now you just nee to know that f ( ) means integrate the function f(). You can also practise integration using the interactive questions Inefinite integration of polynomials. Note that w.r.t. means with respect to an just means that is the variable that you are using. In these questions, is not always the variable being use, but you integrate in eactly the same way whatever letter is use. You coul also try the Inefinite integrals puzzle. This also uses the notation f ( ) to mean integrate the function f(). The Calculus Basic puzzle inclues both ifferentiation an integration, using positive integer powers of. The Calculus Avance puzzle inclues both ifferentiation an integration, using fractional an negative powers of. Fining the arbitrary constant If you are given aitional information, you can fin the value of the arbitrary constant by substituting the given information. This is sometimes calle fining the particular solution of a ifferential equation. The net eample shows how this is one. Eample The graient of a curve at any point (, y) is given by y = ( + ). The curve passes through the point (, ). Fin the equation of the curve. of /0/ MEI

5 WJEC C Integration Notes an Eamples = ( + ) = + Integrating: y c c Just as with ifferentiating, you nee to epan the brackets first When =, y = = c c = = So the equation of the curve is y = Substitute the given values of an y You can see more eamples like the one above using the Flash resource Reverse of ifferentiation. Eample The graient function of a curve is given by an the curve passes through the point (, 9) Fin the equation of the curve. y c c When =, y = c c 9 c Integrate to fin an epression for y in terms of Substitute the coorinates of the given point to fin the value of the constant c. The equation of the curve is y of /0/ MEI