INTEGRATION AND PRIMITIVE ESSENTIALS

INTRODUCTION

Integrals together with derivatives are fundamental objects in Calculus; a very clear conceptual understanding of these is a must. This chapter summarizes the principles of integration and the link between integrals and primitives.

DEFINITE INTEGRAL - DEFINITION

Let f denote a continuous and positive function on an interval [a, b].

By definition the Definite Integral of f between a and b, denoted by ∫ab f   or  ∫ab f(t)dt,  is the area between the f curve and the abscissa axis, delimited by a and b.

It is called a “definite” integral because of its dependence on the two given constants a and b.

The concept can be extended to a non-positive function, bearing in mind that areas in the negative portions of the function are negative.   

INDEFINITE INTEGRAL or PRIMITIVE

The Indefinite Integral or Primitive is a generalization of the Definite Integral. It is a function (as opposed to a definite value) depending on a variable, say x, which replaces the constant value b.

Notation: F(x) = ∫ax f   also written as ∫ax f(t)dt

KEY PROPERTIES OF PRIMITIVES

  • F’(x) = f(x):  given that F(x) = ∫ax f , then the derivative of F is f ; the primitive can be looked at as the “inverse” of the derivative.
  • If F is a primitive of f then F plus any constant is also a primitive of f, since the derivative of a constant is 0; so there is an infinite number of primitives of a given function f, all differing by a constant term.
  • ∫ab f = F(b) – F(a): formula to calculate a Definite Integral as the difference of the primitive at two given points b and a.

IN SUMMARY

EXAMPLE APPLICATIONS

  • Direct calculation of ∫ab f for a given function f

∫ab f = F(b) – F(a) where F is the primitive of f .

It is then just a matter of identifying F given the function f, based on the knowledge of derivatives of common functions provided one of them is applicable.

Simple example:

f(x) = x ; find ∫ab f   = ∫ab x   

Derivative of x2  is  2x and derivative of  x2/2   is  x

Therefore the primitive de f(x) is F(x) = x2/2  and   ∫ab f = b2/2 – a2/2

 

  • INTEGRATION BY PARTS: a useful technique for finding the integral of a function when expressed as a product uv’ where v’ is a derivative of which we know the primitive.

Formula:   ∫ uv’= uv - ∫ u’v

Example 1:  ∫ x cos(x)

u = x  ; v’ = cos(x)

u’ = 1  ; v = sin(x)

⇒ ∫ x cos(x) = x sin(x) - ∫ sin(x) = x sin(x) + cos(x) + constant

Example 2:  ∫ x ln(x)

u = ln(x) ;  v’ = x

u’ = 1/x  ; v = x2/2

⇒ ∫ x ln(x) = x2 ln(x)/2 - ∫(1/x) (x2/2) = x2 ln(x)/2  - ∫ x/2   = x2 ln(x)/2   - x2/4  + constant

 

  • INTEGRATION BY SUBSTITUTION: a useful technique for finding the integral of a function when expressed as a product of a composite function g ∘ f (x) = g [f(x)] and of the derivative of f.

Formula:   ∫ g [ f(x)] f’(x) dx = ∫ g(y) dy 

after substituting    f(x) for y   and  f’(x) dx for dy.

Example: ∫ sin(√x) / √x dx

y = √x

dy = 1 / 2√x dx

⇒ ∫ sin(√x) / √x dx =  ∫ sin(y) * (2 dy) = 2 ∫ sin(y) dy  = -2 cos(y) = -2 cos(√x)  + constant