In calculus, the concept of antiderivatives plays a crucial role in solving various problems. An antiderivative, also known as an indefinite integral, is the reverse process of differentiation. It helps us find the original function when we know its derivative.

To find an antiderivative, we can use the Fundamental Theorem of Calculus. This theorem states that if a function f(x) is continuous on a closed interval [a, b] and F(x) is an antiderivative of f(x) on that interval, then the definite integral of f(x) from a to b is equal to F(b) - F(a).

Let's consider an example to understand this better.

Example:

Find the antiderivative of the function f(x) = 3x^2 + 2x.

Solution:

To find the antiderivative, we can use the power rule of integration. The power rule states that for any constant 'n,' the antiderivative of x^n is (1/(n+1)) * x^(n+1).

Applying the power rule, the antiderivative of 3x^2 is (1/3) * x^3, and the antiderivative of 2x is x^2.

Therefore, the antiderivative of f(x) = 3x^2 + 2x is F(x) = (1/3) * x^3 + x^2 + C, where C is the constant of integration.

By finding the antiderivative, we can find the original function that gives rise to the given derivative.

Remember, practice makes perfect, so keep honing your skills in finding antiderivatives using the Fundamental Theorem of Calculus!