Integration

Defining Integration

  • Integration is one of the two main operations in calculus, the other being differentiation. It is the reverse process of differentiation.
  • The integral of a function is a measure of the area under the curve of this function.

Fundamental Theorem of Calculus

  • This theorem connects differentiation and integration. It states that an indefinite integral of a function can be reversed by differentiation.
  • Given a function f which is continuous on the interval [a,b] and F is an antiderivative of f on [a,b]. Then the definite integration from a to b of f(x) dx equals to F(b) - F(a).

Indefinite Integrals

  • An indefinite integral of a function f, denoted ∫f(x) dx, is the most general antiderivative of f in terms of x.
  • Unlike the definite integral, the indefinite integral has the symbol ∫ without limits and is equal to a family of functions (antiderivatives) plus a constant, denoted by ‘+c’.

Fundamental Integration Rules

  • Constants: The integral of a constant k with respect to x is kx + c.
  • Power Rule: The integral of x^n with respect to x is (1/(n+1))x^(n+1) + c, where n ≠ -1.
  • Exponential Functions: The integral of e^x with respect to x is e^x + c.

Integration by Substitution (u-substitution)

  • This is a method used to find integrals of complicated functions. It involves substituting a part of the function (u) to simplify it.
  • We typically try to substitute a function that, when differentiated, is still present somewhere in the integrand.

Integration by Parts

  • This method allows you to integrate products of two functions. It’s essentially the integration version of the differentiation product rule.
  • The formula: ∫udv = uv - ∫vdu, where u and v are functions of x.

Definite Integrals

  • A definite integral computes the difference between the antiderivative at the upper and lower limits of integration.
  • It is denoted as ∫ from a to b [f(x) dx], and it gives a number that represents the signed area under the curve from a to b.

Integrals and Area

  • Integration can be used to calculate areas under curves, volumes of solids of revolution and solve problems involving rates.