## 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 ∫_{a}^{b} f or ∫_{a}^{b} 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) = ∫_{a}^{x} f also written as ∫_{a}^{x} f(t)dt

### KEY PROPERTIES OF PRIMITIVES

- F’(x) = f(x): given that F(x) = ∫
_{a}^{x}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.
- ∫
_{a}^{b}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 ∫
_{a}^{b}f for a given function f

∫_{a}^{b} 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 ∫_{a}^{b} f = ∫_{a}^{b} x

Derivative of x^{2} is 2x and derivative of x^{2}/2 is x

Therefore the primitive de f(x) is F(x) = x^{2}/2 and ∫_{a}^{b} f = b^{2}/2 – a^{2}/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 = x^{2}/2

⇒ ∫ x ln(x) = x^{2 }ln(x)/2 - ∫(1/x) (x^{2}/2) = x^{2 }ln(x)/2 - ∫ x/2 = x^{2 }ln(x)/2 - x^{2}/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

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