082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

13.2 Elementary integration 429
13.2 ELEMENTARYINTEGRATION
Consider the following problem: given dy = 2x, findy(x). Differentiation of the functiony(x)
= x 2 +c, wherecis a constant, yields dy = 2x for anyc. Thereforey(x) =
dx
dx
x 2 +cisasolutiontotheproblem.Asccanbeanyconstant,thereareaninfinitenumber
of different solutions. The constantcis known as aconstant of integration. Inthis example,thefunctionyhasbeenfoundfromaknowledgeofitsderivative.
Wesay2xhas
beenintegrated,yieldingx 2 +c.Toindicatetheprocessofintegrationthesymbols ∫ and
dx are used. The ∫ sign denotes that integration is to be performed and the dx indicates
thatxisthe independent variable. Returning tothe previous problem, we write
dy
dx = 2x
∫
y = 2xdx=x 2 +c
↑ ↑ ↑
symbols for
integration
constant of integration
Ingeneral, if
dy
dx =f(x)
then
∫
y = f(x)dx
Consider a simple example.
Example13.1 Given dy
dx =cosx−x,findy.
Solution We need to find a function which, when differentiated, yields cosx −x. Differentiating
sinx yields cosx, while differentiating −x 2 /2 yields −x. Hence,
∫
y =
(cosx−x)dx=sinx− x2
2 +c
wherecis the constant of integration. Usually brackets are not used and the integral is
written simplyas ∫ cosx−xdx.
Thefunctiontobeintegratedisknownastheintegrand.InExample13.1theintegrand
iscosx−x.