22. Riemann Sums, Integrals and the FTC

c. Indefinite Integrals

The Fundamental Theorem of Calculus tells us that \(\displaystyle I_a(x)=\int_a^x f(x)\,dx\) is a particular antiderivative of \(f(x)\). Each choice of \(a\) gives us a different antiderivative. If we don't specify the \(a\), we get the general antiderivative. This motivates the following definition:

The general antiderivative of a function \(f(x)\) is also called the indefinite integral of \(f(x)\) and is denoted by \[ \int f(x)\,dx \] with no limits on the integral sign. Thus if \(F(x)\) is any particular antiderivative of \(f(x)\), then \[ \int f(x)\,dx=F(x)+C \] where \(C\) is an arbitrary, unspecified constant, called the constant of integration.

The constant of integration can be any letter. \(C\) is just conventional.

For example, since \(\dfrac{d}{dx}(x^2-2x)=2x-2\), we can write \[ \int (2x-2)\,dx=x^2-2x+C \] However, since (by the chain rule) we also have \(\dfrac{d}{dx}(x-1)^2=2(x-1)=2x-2\), we can also write \[ \int (2x-2)\,dx=(x-1)^2+C \] There is no contradiction here because \(x^2-2x\) and \((x-1)^2=x^2-2x+4\) just differ by a constant, so that the two formulas for the indefinite integral just have a different meaning for \(C\).