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Antiderivatives and the Main Theorem

The fundamental theorem of calculus reveals that integration and differentiation are inverse operations. This page follows Walz, Foundations of Mathematics, the middle of Section 6.3: it introduces the integral function, shows that differentiating it recovers the integrand, and uses this to turn the evaluation of definite integrals into the search for antiderivatives.

It continues directly from The Definite Integral, where integrability and the basic properties were established.

The Integral Function

Fixing the lower bound $a$ and letting the upper bound vary defines the integral function $F_a(t) = \int_a^t f(x)\,dx.$

For the power function $f(x) = x^i$ this is $F_a(t) = \dfrac{t^{i+1}}{i+1} - \dfrac{a^{i+1}}{i+1}$, and differentiating with respect to $t$ gives back the integrand: $F_a'(t) = t^i = f(t).$

This is not a coincidence. It holds for every continuous function.

Differentiability of the Integral Function

For a general continuous $f$, the difference quotient of $F_a$ simplifies, using the interval properties of the integral, to $\frac{F_a(t + h) - F_a(t)}{h} = \frac{1}{h}\int_t^{t+h} f(x)\,dx.$

The mean value theorem of integral calculus states that for continuous $f$ there is a point $\xi \in \lbrack t, t+h \rbrack$ with $\int_t^{t+h} f(x)\,dx = f(\xi)\,h$. Hence the difference quotient equals $f(\xi)$, and as $h \to 0$ the point $\xi$ is squeezed towards $t$, giving $f(\xi) \to f(t)$.

Differentiability of the integral function. For any continuous function $f$ on $\lbrack a, b \rbrack$, the integral function $F_a(t) = \int_a^t f(x)\,dx$ is differentiable and $F_a'(t) = f(t).$

So differentiating cancels integrating; conversely $\int_a^t F_a'(x)\,dx = F_a(t)$, so integrating cancels differentiating. This is the sense in which the two operations are inverse.

Antiderivatives (Primitives)

The integral function $F_a$ is defined only implicitly, as an integral. To evaluate integrals we look instead for an explicit function whose derivative is $f$.

Antiderivative. A differentiable function $F$ with $F' = f$ is called an antiderivative (or primitive) of $f$.

Every derivative rule read backwards supplies antiderivatives. For example:

  • \[F(x) = \dfrac{x^{i+1}}{i+1}\]

    is an antiderivative of $x^i$.

  • \[F(x) = e^x\]

    is an antiderivative of $e^x$; the natural logarithm $\ln(x)$ is an antiderivative of $\tfrac{1}{x}$ for $x > 0$.

  • \[F(x) = -\cos(x)\]

    is an antiderivative of $\sin(x)$.

Antiderivatives are not unique, but they are unique up to a constant:

Antiderivatives differ only by a constant. If $F$ and $G$ are both antiderivatives of $f$, then $F(x) = G(x) + c$ for some constant $c$, because $(F - G)' = f - f = 0$.

The Main Theorem

Since the integral function $F_a$ is itself an antiderivative of $f$, any antiderivative $F$ differs from it only by a constant. Evaluating that constant at $t = a$ yields $F_a(t) = F(t) - F(a)$, and setting $t = b$ gives the central result.

Main theorem of differential and integral calculus. If $f$ is continuous on $\lbrack a, b \rbrack$ and $F$ is any antiderivative of $f$, then $\int_a^b f(x)\,dx = F(b) - F(a).$

The evaluation is written with the bracket notation $\bigl\lbrack F(x) \bigr\rbrack_a^b = F(b) - F(a).$

The machinery for evaluating a definite integral is therefore:

  1. Find any antiderivative $F$ of $f$.
  2. Compute $F(b)$ and $F(a)$ by substituting the bounds.
  3. The difference $F(b) - F(a)$ is the definite integral.

Worked Examples

  • With $F(x) = \tfrac{1}{3}x^3$ an antiderivative of $x^2$: $\int_1^{10} x^2\,dx = \frac{10^3}{3} - \frac{1^3}{3} = \frac{999}{3} = 333.$ Using $G(x) = \tfrac{1}{3}x^3 + 15$ instead gives the same result — the additive constant cancels.
  • Since $(e^x)' = e^x$: $\int_1^{3} e^x\,dx = \bigl\lbrack e^x \bigr\rbrack_1^3 = e^3 - e = e(e^2 - 1).$
  • Since $(-\cos(x))' = \sin(x)$: $\int_0^{\pi} \sin(x)\,dx = \bigl\lbrack -\cos(x) \bigr\rbrack_0^\pi = -\cos(\pi) + \cos(0) = 2.$

Where This Leads

The main theorem reduces integration to finding antiderivatives — but for many integrands no antiderivative is obvious. The next page, Integration Techniques, develops partial integration and the substitution rule, which produce antiderivatives for many further functions.