60ReviewIntegration

= Antidifferentiation (integration) =


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


 * Recall **

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

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

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

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


 * Notation **

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

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


 * Polynomials **

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

Examples


 * Integration of (ax + b) n where n ** **¹** ** –1 **


 * Recall that by the chain rule for differentiation:

math . \qquad \dfrac{d}{dx} \big( ax+b \big)^{n+1} = a(n+1)(ax+b)^n \qquad. math


 * Therefore, the reverse is true

math , \qquad \displaystyle{ \int {a(n+1)(ax+b)^n} \; dx} = \big( ax+b \big)^{n+1} + c \qquad. math


 * This can be simplified by moving the constants out of the integral:

math . \qquad a(n+1) \displaystyle { \int {(ax+b)^n}\; dx} = \big( ax+b \big)^{n+1} + c \qquad. math


 * Therefore:

math . \qquad \displaystyle{ \int {(ax+b)^n}\; dx} = \dfrac{ \big( ax+b \big)^{n+1}}{a(n+1)} + c_1 \qquad. math

Note: This only works for a __linear__ term raised to a power.

Note: For Specialist students, it is preferred that you use Integration by Substitution rather than this rule.

Examples

Integration into Natural Logarithms

math . \qquad \text{Since: } \; \dfrac {d}{dx} \Big( \log_e (x) \Big) = \dfrac {1}{x} \qquad. math

math . \qquad \text{Then: } \; \displaystyle{\int} \dfrac{1}{x} \; dx = \log_e |x| + c, \quad \text { where } \; x \neq 0 \qquad. math

math . \qquad \text {Also since: } \; \dfrac {d}{dx} \Big( \log_e \big( g(x) \big) \Big) = \dfrac {g'(x)}{g(x)} \qquad. math

math . \qquad \text{Then: } \; \displaystyle{\int} \dfrac{g'(x)}{g(x)} \; dx = \log_e |g(x)| + c, \quad \text { where } \; g(x) \neq 0 \qquad. math

Examples

Using the calculator to find integrals.

.