Integration by Substitution

From the Endpoint Formula to Substitution

The Fundamental Theorem of Calculus, Part II, evaluates an integral by finding an antiderivative. Substitution extends that method: rather than searching for an antiderivative in the original variable, we recognize the integrand as the derivative of a composite function. The chain rule explains why a derivative factor appears, and the endpoint formula determines the transformed limits.

We will use the convention that reversing the endpoints reverses the sign of an integral: \(\int_v^u f=-\int_u^v f\), and \(\int_u^u f=0\). These conventions matter when the change of variable is decreasing, or when its endpoint values are equal. In particular, the substitution formula is an identity for signed integrals; it does not require the change of variable to be increasing.

For this tutorial, a function \(\phi\) is continuously differentiable on \([a,b]\) if it is continuous on \([a,b]\), differentiable on \((a,b)\), and its derivative is continuous there and extends continuously to the endpoints. The proof below needs differentiability only at interior points. We impose a slightly generous domain condition on the other function so that the chain rule applies even when \(\phi(x)\) is an endpoint of its range.

The Substitution Theorem

Proof. Choose any \(u_0\in J\), and define \(G:J\to\mathbb{R}\) by

Because \(f\) is continuous on the open interval \(J\), the Fundamental Theorem of Calculus, Part I, gives \(G'(u)=f(u)\) at every \(u\in J\). The function \(G\circ\phi\) is continuous on \([a,b]\). At every interior point \(x\in(a,b)\), the value \(\phi(x)\) belongs to \(J\), where \(G\) is differentiable. The chain rule therefore gives

The function on the right is continuous on \([a,b]\), since \(f\), \(\phi\), and \(\phi'\) are continuous there. Thus FTC Part II applies to \(G\circ\phi\) and yields

By the definition of \(G\) and additivity of the integral across adjacent intervals, the difference on the right is the oriented integral from \(\phi(a)\) to \(\phi(b)\):

Combining the equalities proves the formula. The domain assumption on \(J\) ensures that \(G\) is differentiable at every value \(\phi(x)\), including values that may be endpoints of the range of \(\phi\). \(\square\)

The theorem does not require \(\phi\) to be one-to-one or monotone. It also allows \(\phi'\) to vanish. Those features are not accidental: the proof uses the chain rule and the endpoint formula, not an inverse function for \(\phi\).

How to Choose the Pieces of a Substitution

A useful way to apply the theorem is to match the integrand with \(f(\phi(x))\phi'(x)\). The transformed integral has endpoints \(\phi(a)\) and \(\phi(b)\), in that order. A substitution calculation therefore has three linked parts:

This is more than a convenient notation for “letting \(u=\phi(x)\).” The theorem verifies that the transformed integrand is continuous and that the integral identity is valid. In particular, the factor \(\phi'(x)\) cannot be dropped: it records how the change in the inner variable relates to a change in \(x\).

Worked Applications

Applying Substitutions in Succession

Sometimes an integral is easiest to transform in two stages. The next proposition makes precise that successive substitutions agree with their composite. Its domain conditions ensure that each use of the substitution theorem is legitimate.

Proof. First apply the substitution theorem to \(\psi\) on the interval with endpoints \(\phi(a)\) and \(\phi(b)\), using \(f\) as the outer integrand. The interval between these endpoints lies in \(\phi([a,b])\), since the continuous image \(\phi([a,b])\) is an interval. The domain conditions give

Now define \(h(t)=f(\psi(t))\psi'(t)\) for \(t\in J\). The assumptions ensure that \(h\) is continuous on \(J\). Apply the substitution theorem to \(\phi\) and \(h\):

By the definition of \(h\), the right-hand side is

Combining the two applications proves the proposition. \(\square\)

The derivative factors multiply because the chain rule gives \((\psi\circ\phi)'(x)=\psi'(\phi(x))\phi'(x)\). The proposition is useful when a complicated inner expression is naturally built in stages. It also shows why transforming the limits and transforming the derivative factor are parts of the same operation.

Common Pitfalls and the Role of the Hypotheses

A frequent error is to change the integrand but leave the original limits in place. The limits must be evaluated under the same function used in the integrand: if \(u=\phi(x)\), then the endpoints become \(\phi(a)\) and \(\phi(b)\), in that order. The decreasing example shows why order matters.

Another pitfall is to assume that substitution requires an inverse function. The theorem proved here does not: \(\phi\) may fail to be one-to-one, and it may have interior maxima, minima, or points where \(\phi'=0\). The nonmonotone example is valid precisely because the proof works with an antiderivative of \(f\) and the chain rule, rather than trying to invert \(\phi\).

The open-interval condition on \(f\) is also useful, not merely technical decoration. FTC Part I gives an ordinary derivative for the antiderivative at every point of that open interval. Since every value \(\phi(x)\) lies inside it, the chain rule applies at every interior \(x\), including one where \(\phi(x)\) is an endpoint of the range \(\phi([a,b])\). Taking \(f\) only on the closed range could leave that differentiability step unjustified at such a value.

Finally, substitution for signed integrals includes the factor \(\phi'(x)\), not its absolute value. An absolute value may arise in other settings involving lengths or areas, but it is not part of this Riemann integral identity. Here the sign of \(\phi'\), together with the order of the transformed endpoints, records orientation.