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 .