The *fundamental theorem of calculus* roughly says that integration and differentiation are inverses of each other. It has two parts.

#### First fundamental theorem of calculus

If we integrate and then differentiate, we get back to where we started.

More precisely, if \(f(x)\) is a continuous function and \(a\) is a constant, then the function \(F(x)\) defined by \[F(x)=\int_a^x f(x)\,dx\] is differentiable, with \(F'(x)=f(x)\).

#### Second fundamental theorem of calculus

If we differentiate and then integrate, we get back to where we started, up to a constant.

More precisely, if \(F(x)\) has a continuous derivative \(f(x)=F'(x)\), then \[\int_a^b f(x)\,dx=F(b)-F(a).\]

The fundamental theorem is regularly used in order to calculate integrals. For example, the derivative of \(\frac{1}{3}x^3\) is \(x^2\), so \[\int_0^3 x^2\,dx=\bigl[\tfrac{1}{3}x^3\bigr]_0^3= \tfrac{1}{3}(3^3)-\tfrac{1}{3}(0^3)=9.\]

There are also stronger versions of these theorems.