The Fundamental Theorem of Calculus

In the previous post, we defined two different kinds of integrals: the indefinite integral and the definite integral. The indefinite integral derives its meaning from being the inverse process of differentiation, while the definite integral in itself has the geometric meaning of the area under a curve. The fundamental theorem of calculus shows that in a certain sense these two are entirely the same process, and one could even say that calculus itself was born from this.

The Fundamental Theorem of Calculus

Since the indefinite integral is by its very definition the inverse process of differentiation, what is meaningful in this process is that the definite integral actually yields a result similar to the inverse process of differentiation. However, because the definite integral is ultimately a tool that outputs a value, its resulting output is not a function. Accordingly, we take the upper limit of the definite integral as a variable and define the following function:

\[\int_a^x f(t)\dd{t}\]

Here \(t\) is simply a dummy variable, introduced to avoid confusion with the \(x\) (already) placed in the upper limit; if there is no risk of confusion, one may just as well write \(x\) in place of \(t\).

Theorem 1 (Fundamental theorem of calculus) If \(f\) is continuous on \([a,b]\) and we define \(F(x) = \int_a^x f(t)\dd{t}\), then \(F\) is differentiable on \([a,b]\) and at every \(x\),

\[F'(x) = f(x)\]

holds. Here differentiability at the endpoints \(a, b\) means the right-hand and left-hand derivatives of §Differentiation and Derivatives, ⁋Definition 6.

Proof

By the second result of §Integration, ⁋Proposition 11, for \(h>0\),

\[F(x+h) - F(x) = \int_a^{x+h} f - \int_a^x f = \int_x^{x+h} f(t)\dd{t}\]

holds. Then, since \(f\) is continuous on \([x, x+h]\), by §Continuous Functions, ⁋Theorem 4 (Extreme Value Theorem) it attains a minimum \(m_h\) and a maximum \(M_h\) on that interval, and by the third result of §Integration, ⁋Proposition 11,

\[m_h \leq \frac{F(x+h) - F(x)}{h} \leq M_h\]

holds. Now, as \(h \rightarrow 0\), the interval \([x, x+h]\) shrinks to a single point \(x\), and by the continuity of \(f\) we have \(m_h, M_h \rightarrow f(x)\); hence by §Limits of Functions, ⁋Proposition 8 (Squeeze Theorem), the average rate of change converges to \(f(x)\). For the case \(h < 0\), the proof can be completed in a similar manner.

This theorem is a rigorous expression of the intuition that the rate at which area accumulates is precisely the height. That is, for the function \(F(x)\) defined by the integral, its rate of increase \(F'(x)\) at a point is precisely the value \(f(x)\) of the integrand at that point.

Corollary 2 Every function continuous on \([a,b]\) has a primitive. Specifically, \(F(x) = \int_a^x f(t)\dd{t}\) is one such primitive.

The function \(F(x)\) defined above fixes the integration constant \(C\) to a single value; concretely, one may think of it as choosing the constant of integration so that \(F(a)=0\).

On the other hand, the above corollary is a different matter from the statement that the integral of an arbitrary function can be expressed in terms of elementary functions; it asserts only that \(\int_a^x f(t)\dd{t}\) defined by the above formula is in itself a primitive of \(f\). For instance, consider the following example.

Example 3 (Derivative of a special function) The error function is defined by

\[\erf(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\dd{t}\]

Since the primitive of \(e^{-t^2}\) cannot be written as an elementary function, this integral has no closed form, but by Theorem 1 (Fundamental theorem of calculus), \(\erf\) is differentiable and

\[\erf'(x) = \frac{2}{\sqrt{\pi}}e^{-x^2}\]

Likewise, for the logarithmic integral \(\mathrm{Li}(x) = \int_2^x \dd{t}/\ln t\) as well, the derivative immediately follows as \(\mathrm{Li}'(x) = 1/\ln x\) for \(x > 1\). In this way, a function defined by an integral becomes an object of calculus in its own right.

A slightly more advanced form of expression will be introduced again after examining the integration of power series in Proposition 7 (Term-by-term integration of power series).

Meanwhile, combining Theorem 1 (Fundamental theorem of calculus) with the fact that the only functions whose derivative is \(0\) are constant functions yields a powerful tool for computing definite integrals as the difference of primitives.

Theorem 4 If \(f\) is continuous on \([a,b]\) and \(G\) is any primitive of \(f\), then

\[\int_a^b f(x)\dd{x} = G(b) - G(a)\]

Proof

Setting \(F(x) = \int_a^x f\), by Theorem 1 (Fundamental theorem of calculus), \(F\) is also a primitive of \(f\). Since two primitives differ only by a constant (§Mean Value Theorem, ⁋Corollary 5), there exists a constant \(C\) such that \(F = G + C\). Since \(F(a) = \int_a^a f = 0\), we have \(C = -G(a)\), and therefore

\[\int_a^b f = F(b) = G(b) + C = G(b) - G(a)\]

One commonly writes \(G(b) - G(a)\) as \(\bigl[G(x)\bigr]_a^b\). Thanks to this theorem, the computation of a definite integral is reduced to the problem of finding a primitive rather than taking a limit of Riemann sums. For instance, \(\int_0^1 x^2\dd{x} = \bigl[x^3/3\bigr]_0^1 = 1/3\) agrees with the value obtained laboriously via Riemann sums in §Integration, but here it is obtained by an entirely different calculation, namely substituting the endpoints into the primitive.

In particular, Theorem 1 (Fundamental theorem of calculus) combines with §Differentiation, ⁋Theorem 4 (Chain rule) when the upper and lower limits of integration depend on a variable. If the upper limit is a function \(g(x)\), then setting \(F(u) = \int_a^u f\), we have \(\int_a^{g(x)} f = F(g(x))\), and since \(F'(u) = f(u)\), the chain rule gives \(\frac{d}{\dd{x}}\int_a^{g(x)} f(t)\dd{t} = f(g(x))g'(x)\). If both limits are variable, one splits the interval into two parts and applies the rule to each side.

Proposition 5 (Leibniz rule) If \(f\) is continuous on an interval \(I\), and \(g, h\) are differentiable with values in \(I\), then

\[\frac{d}{\dd{x}}\int_{h(x)}^{g(x)} f(t)\dd{t} = f(g(x))g'(x) - f(h(x))h'(x)\]

Proof

Pick a point \(c\) in \(I\) and split the integral into two parts using the additivity over intervals from §Integration, ⁋Proposition 11:

\[\int_{h(x)}^{g(x)} f(t)\dd{t} = \int_c^{g(x)} f(t)\dd{t} - \int_c^{h(x)} f(t)\dd{t}\]

Setting \(F(u) = \int_c^u f(v)\dd{v}\), we have \(F'(u) = f(u)\) by Theorem 1 (Fundamental theorem of calculus), and the right-hand side is \(F(g(x)) - F(h(x))\). Applying the chain rule to each term gives

\[\begin{aligned} \frac{d}{\dd{x}}\bigl[F(g(x)) - F(h(x))\bigr] &= F'(g(x))g'(x) - F'(h(x))h'(x) \\[2pt] &= f(g(x))g'(x) - f(h(x))h'(x) \end{aligned}\]

However, caution is required when the integrand is not defined or is discontinuous on the interval of integration. Consider the following example.

Example 6 For the integral

\[\int_{-1}^{1} \frac{\dd{x}}{x^2}\]

formally substituting

\[\bigl[-x^{-1}\bigr]_{-1}^{1} = -1 - 1 = -2\]

yields a negative number, but since the integrand \(1/x^2\) is always positive, this is clearly an incorrect calculation. The reason this is incorrect is that the integrand diverges at \(x = 0\) and is therefore not continuous on \([-1,1]\). Since the hypothesis of Theorem 4 is violated, one cannot apply the theorem as is; in fact, this integral diverges.

Term-by-Term Integration of Power Series

On the other hand, integration also works well with power series. The following proposition likewise requires knowledge of analysis for its proof, so we shall accept it as fact for now.

Proposition 7 (Term-by-term integration of power series) If \(f(x) = \sum_{n=0}^\infty c_n x^n\) has radius of convergence \(R > 0\), then for \(\lvert x\rvert < R\),

\[\int_0^x f(t)\dd{t} = \sum_{n=0}^\infty \frac{c_n}{n+1} x^{n+1}\]

and this series also has radius of convergence \(R\).

Then we can see in what form \(e^{-x^2}\) examined earlier in Example 3 (Derivative of a special function) can be written.

Example 8 Let us return to the error function of Example 3 (Derivative of a special function). From the power series of the exponential function,

\[e^{-t^2} = \sum_{n=0}^\infty \frac{(-1)^n}{n!}t^{2n}\]

and its radius of convergence is \(\infty\). Now term-by-term integration via Proposition 7 (Term-by-term integration of power series) gives

\[\int_0^x e^{-t^2}\dd{t} = \sum_{n=0}^\infty \frac{(-1)^n}{n!(2n+1)}x^{2n+1}\]

and therefore

\[\erf(x) = \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infty \frac{(-1)^n}{n!(2n+1)}x^{2n+1}\]

That is, although the primitive still cannot be written as an elementary function, it can be expressed explicitly as a power series in this way.

By a similar method, term-by-term integration of \(1/(1 + x^2)\) and \(1/(1+x)\) yields, within their respective radii of convergence,

\[\arctan x = \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} x^{2n+1}\]

and

\[\ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} x^n\]

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References

[Ste] J. Stewart, Calculus, 8th ed., Cengage Learning, 2016.
[Kim] 김홍종, 미적분학 1·2, 제3개정판, 서울대학교출판문화원, 2020.