Integral Calculus

Introduction

In this section we will review the defintions and rules of integration, and look at some simple examples

Definite Integrals

The definite integral of a function \(f(x)\) over the interval \([a,b]\) is (usually) defined as the limit of a sum:

\[ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x \]

where \(x_i^*\) is any point in the interval \([a+(i-1)\Delta x, a+i\Delta x]\) and \(\Delta x = \frac{b-a}{n}\).

The definite integral is thus the area under the curve \(f(x)\) between \(x=a\) and \(x=b\).

Indefinite Integrals

The indefinite integral of a function \(f(x)\) is a function (the anti-derivative) \(F(x)\) such that \(F'(x) = f(x)\) (i.e. the inverse of differentiation).

Since the derivative of a constant is zero, the indefinite integral of a function is only defined up to a constant. For example, the indefinite integral of \(f(x) = x^2\) is \(F(x) = \frac{1}{3}x^3 + c\). The constant \(c\) is called the constant of integration.

Rules of Integration

Linearity

Integration is a linear operation, i.e. for functions \(f(x)\) and \(g(x)\) and constants \(a\) and \(b\),

\[ \int (af(x) + bg(x)) dx = a \int f(x) dx + b \int g(x) dx \]

This is the equivalent of the sum rule for differentiation.

Integration by parts

The integration by parts formula is

\[ \int u dv = uv - \int v du \]

where \(u\) and \(v\) are functions of \(x\). This is a reaarrangement of the product rule for differentiation.

Substitution/Change of variable

The substitution rule is

\[ \int f(g(x)) g'(x) dx = \int f(u) du \]

where \(u = g(x)\).

This is the equivalent of the chain rule for differentiation.

Examples

Polynomial functions

The indefinite integral of a polynomial function of the form

\[ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \dots + a_n x^n \]

is given by

\[ \int f(x) dx = a_0 x + \frac{a_1}{2} x^2 + \frac{a_2}{3} x^3 + \frac{a_3}{4} x^4 + \dots + \frac{a_n}{n+1} x^{n+1} + c \]

where \(c\) is an arbitrary constant.

More generally, the function

\[ \int x^n dx = \frac{x^{n+1}}{n+1} + c \]

applies for any real number \(n\neq -1\). In that special case, we have

\[ \int \frac{1}{x} dx = \ln |x| + c \]

Proving this is easiest by starting (in reverse with the chain rule, applied to the function \(g(x) = \exp(\ln x) \):

\(\frac{dg}{dx} = \frac{d}{dx} \exp(\ln x) = \exp(\ln x) \frac{d}{dx} \ln x = x \frac{d}{dx} \ln x \).

But \( \exp(\ln x) = x\), so \(\frac{dg}{dx} = 1 = x \frac{d}{dx} \ln x \), and hence \(\frac{d}{dx} \ln x = \frac{1}{x}\).

Trigonometric functions

For the primary functions, these can be derived from the derivatives of rules for differentiation in reverse (not forgetting the constant of integration).

Unlike differentiation, inverse trignometric functions sometimes appear on formula sheets for integration. For example

\[ \int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x + c, \]

\[ \int \frac{1}{1+x^2} dx = \arctan x + c. \]

These results can generally be proved by appropriate sustitutions, and use of trignometric identities, e.g. setting \(x=\sin t\) and using the identity \(\sin^2 t + \cos^2 t = 1\).