10Aantidifferentiation

= Antidifferentiation (Integration) =


 * This is the reverse of differentiation.
 * Given a gradient function, it allows us to find the original function.


 * ** Antidifferentiation ** is also called ** Integration **
 * To __**Antidifferentiate**__ a function is the same as to **__Integrate__** a function


 * Recall **

math . \qquad \dfrac{d}{dx} \Big( f(x) \Big) \; \text { means differentiate } f(x) \text { with respect to } x math

math . \qquad \dfrac{d}{dx} \Big( f(x) \Big) = f '(x) math

... ... so //**f(x)**// is the antiderivative of //**f '(x)**//

math . \qquad f(x) = \displaystyle{ \int{ f'(x) } \; dx + c } math


 * Notation **

math . \qquad \displaystyle{ \int } \; \text { means antidifferentiate (or integrate)} math

... ... //**dx**// means the integration is with respect to the variable **//x//**

math . \qquad \displaystyle{\int } \;\; dx \qquad \text { the two symbols together act as brackets -- everything inside is integrated} math

... ... The **dx** symbol MUST be included every time!!

... .... Sometimes we use **//F(x)//** as the antiderivative of **//f(x)//**

math . \qquad F(x) + c = \displaystyle{ \int{ f(x) } \; dx } + c math


 * Constant of Antidifferentiation (c) **


 * Remember that when we differentiate, any constant term is lost.


 * So when we antidifferentiate we cannot know the value of the constant term (unless given extra information).


 * Therefore we indicate the unknown constant term with the ** constant of antidifferentiation **(usually c).


 * You should (almost) always write the + c when antidifferentiating.


 * There are only two circumstances when you don't need to write + c
 * 1) When the question asks you to find __an__ antiderivative.
 * 2) In that case, c = 0 is an acceptable solution so write the antiderivative without the + c.
 * 3) It would not be wrong to write + c in these questions so if in doubt, write + c.
 * 4) When calculating Definite Integrals, the c term cancels out during the calculations so it is not needed.


 * Properties of Integrals **

math . \qquad \displaystyle{ \int{ f(x) + g(x) } \; dx} = \displaystyle{ \int{ f(x) } \; dx + \int{ g(x) } \; dx} \quad \textit { for any functions, f and g} math

math .\qquad \displaystyle{ \int{ kf(x) } \; dx} = \displaystyle{ k \int{ f(x) } \; dx} \quad \textit { for any constant, k} math


 * Polynomials **

math . \qquad \displaystyle{ \int {ax^n} \; dx} = \dfrac {ax^{n+1}}{n+1} + c, \quad n \ne -1 math


 * __**increase**__ the index of each term by one
 * __**divide**__ by the __**new**__ index
 * add **c** at the end


 * The same rule applies when n is fraction or negative


 * Example 1 **

math . \qquad \text{Find the antiderivative } \; \displaystyle{ \int \; 8x^3 -x^2 + \dfrac{2x}{3} - 7 \; dx} math


 * Solution:**

math . \qquad \displaystyle{ \int \; 8x^3 -x^2 + \dfrac{2x}{3} - 7 \; dx} \\.\\ . \qquad = \dfrac{8x^4}{4} -\dfrac{x^3}{3} +\dfrac{2x^2}{6} - \dfrac{7x^1}{1} + c \\.\\ . \qquad = 2x^4-\dfrac{x^3}{3} + \dfrac{x^2}{3} - 7x + c math

More Examples

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