Definite Integrals

Definition of a Definite Integral

Let f(x) be a defined integral in an interval a ≤ x ≤ b. Divide the interval into n equal parts of length Δx = (b - a)/n. Then the definite integral of F(x) between x = a and x = b is defined as

The limit will certainly exist if f(x) is piecewise continuous.

If f(x) = (d/dx)g(x), then by the fundamental theorem of the integral calculus the above definite integral an be evaluated by using the result

If the interval is infinite or if f(x) has a singularity at some point in the interval, the definite integral is called an improper integral and can be defined by using appropriate limiting procedures. For example,

General Formulas Involving Definite Integrals

This is called the mean value theorem for definite integrals and is valid if f(x) is continuous in a ≤ x ≤ b.

This is a generalization of the previous one and is valid if f(x) and g(x) are continuous in a ≤ x ≤ b and g(x) ≥ 0.

Leibnitz's Rule for Differentiation of Integrals

Approximate Formulas for Definite Integrals

In the following the interval from x = a to x = b is subdivided into n equal parts by the points a = x₀, x₂, . . ., x_{n - 1}, x_n = b and we let y₀ = f(x₀), y₁ = f(x₁), y₂ = f(x₂), . . ., y_n = f(x_n), h = (b - a)/n.
Rectangular formula

Trapezoidal formula

Simpson’s formula (or parabolic formula) for n even

Definite Integrals Involving Rational or Irrational Expressions

Definite Integrals Involving Trigonometric Functions

All letters are considered positive unless otherwise indicated.

Definite Integrals Involving Exponential Functions

For even n this can be summed in terms of Bernoulli numbers.

For some positive integer values of n the series can be summed.

Definite Integrals Involving Logarithmic Functions

If n ≠ 0, 1, 2, ... replace n! by Γ(n + 1).

Definite Integrals Involving Hyperbolic Functions

If n is an odd positive integer, the series can be summed.

Miscellaneous Definite Integrals

This is called Frullani’s integral. It holds if f’(x) is continuous and converges.