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. Integration as inverse operation of differentiation. If d / dx { φ( x)) = f( x), ∫f( x) dx = φ( x) + C, where C is called the constant of integration or arbitrary constant.
Symbols function f( x) Integrand f( x) dx → Element of integration ∫→ Sign of integral
φ( x) → Anti-derivative or primitive or integral of function f( x) The process of finding functions whose derivative is given, is called antidifferentiation or integration.
Geometrical Interpretation of Indefinite Integral
If d / dx { φ( x)} = f( x), then ∫f( x) dx = φ( x) + C. For different values of C, we get different functions, differing only by a constant. The graphs of these functions give us an infinite family of curves such that at the points on these curves with the same x-coordinate, the tangents are parallel as they have the same slope φ '( x) = f( x).
Consider the integral of 1 / 2√x i. e., ∫1 / 2√xdx = √x + C, C ∈ R
Above figure shows some members of the family of curves given by y = + C for different C ∈ R.