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Integration Rules and Formulas

December 10, 2020 by Prasanna

Integration Rules and Formulas

Integral of a Function

A function ϕ(x) is called a primitive or an antiderivative of a function f(x), if ?'(x) = f(x).
Let f(x) be a function. Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by ∫f(x) dx.
Thus,
Integration Rules and Formulas 1
where ϕ(x) is primitive of f(x) and c is an arbitrary constant known as the constant of integration.

Integration Rules and Formulas 2

Integration Rules

  1. Chain rule :
    ∫u.v dx = uv1 – u’v2 + u”v3 – u”’v4 + ……… + (–1)n­–1 un–1vn + (–1)n ∫un.vn dx
    Where  stands for nth differential coefficient of u and stands for nth integral of v.
  2. Sum Rule
    ∫(f + g) dx = ∫f dx + ∫g dx
  3. Difference Rule
    ∫(f – g) dx = ∫f dx – ∫g dx
  4. Multiplication by constant
    ∫cf(x) dx = c∫f(x) dx
  5. Power Rule (n≠-1)
    ∫xn dx = xn+1/(n+1) + C

Fundamental Integration Formulae

Integration Rules and Formulas 3
Integration Rules and Formulas 4
In any of the fundamental integration formulae, if x is replaced by ax+b, then the same formulae is applicable but we must divide by coefficient of x or derivative of (ax+b) i.e., a. In general, if ∫f(x) dx = ϕ(x) + c, then
Integration Rules and Formulas 5

Some more Results
Integration Rules and Formulas 7

Filed Under: Mathematics Tagged With: Fundamental Integration Formulae, Integral of a Function, Integration Rules

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