What are Antiderivatives?

Antiderivative of a function is the reverse operation of the function’s derivative. Mathematically, we also call Antiderivative the Integral of a function.

Suppose, the derivative of the function F(x) is,

F'(x) = f(x)

For all x in the domain of f. If f(x), is the derivative of F(x) then the antiderivative or integral of the function f(x) is,

This can be explain by the example lets take a function f(x) = x², on differentiating this function, the output is another function g(x) = 2x.

For, g(x) = 2x the anti-derivative will be, 

f(x) = x²

d/dx[f(x)] = f'(x) = g(x)

g(x) = 2x

Now the antiderivative of 2x is,

= ∫g(x).dx

= ∫(2x).dx

= 2(x²)/2 + C

= x² + C

Here the symbol ∫ denotes the anti-derivative operator, it is called indefinite integrals. Also, C is integration constant, or Antiderivative constant. Antiderivatives are classified into two types,

• Indefinite Antiderivatives
• Definite Antiderivatives

Indefinite Antiderivative

Indefinite antiderative also called the indefinite integral is anti derivative of that function in which the limit of the antiderivative (integration) is not given and the result is accompanied with a constant value (generally C) called the constant of integration. Suppose we have a function F(x) whose derivative is f(x) then,

∫ f(x) dx = F(x) + C

where C is Integration Constant

Definite Antiderivative

Definite Antiderivative or definite integral is the antiderivative of any function inside a closed interval. In this integration the constant of integration is not present and the answer to the integration is some contact value. Suppose we have a function F(x) that is defined on closed interval [a, b] then if its derivative is f(x), its definite antiderative is written as,

∫_a^b f(x) = [F(x)]_a^b = F(b) – F(a)

Antiderivatives: The Antiderivative of a function is the inverse of the derivative of the function. Antiderivative is also called the Integral of a function. Suppose the derivative of a function d/dx[f(x)] is F(x) + C then the antiderivative of [F(x) + C] dx of the F(x) + C is f(x). An example explains this if d/dx(sin x) is cos x then, the antiderivative of cos x, given as ∫(cos x) dx is sin x.

Antiderivative of any function is used for various purposes, to give the area of the curve, to find the volume of any 3-D curve, and others. In this article, we will learn about, Antiderivatives, Antiderivatives Formulas, Antiderivatives rules, and others in detail.