Integrating Square Roots of Difference Forms

Why Difference Forms Need Care

After completing the square, a quadratic under a square root may become a constant multiple of either \(u^2-a^2\) or \(a^2-u^2\). These two forms look similar, but their real domains and antiderivatives differ: one is defined outside a bounded interval, while the other is defined inside it.

The key is to keep track of three things at once: the completed-square substitution, the real values for which the radical exists, and the numerical factors introduced by the change of variable. In particular, the antiderivative of \(\sqrt{u^2-a^2}\) contains a factor of one-half. Differentiating the formula is a reliable way to check that factor.

The Two Real Difference Forms

For the first form, the radicand is nonnegative precisely when \(u^2-a^2\ge 0\), or equivalently \(|u|\ge a\). Thus \(\sqrt{u^2-a^2}\) is real on \((-\infty,-a]\) and \([a,\infty)\). Since these are separate intervals, an antiderivative may have a different constant on each interval.

For the second form, \(a^2-u^2\ge 0\) precisely when \(|u|\le a\). The radical \(\sqrt{a^2-u^2}\) is real on \([-a,a]\), and its derivative-based antiderivative formula applies on the interior \((-a,a)\). The formula also extends continuously to the endpoints.

Antiderivative Formulas

On either interval where \(|u|>a\), the antiderivative for the unbounded difference form is

The absolute value is needed because \(u+\sqrt{u^2-a^2}\) is negative when \(u\le -a\). It is nonzero on the domain: for \(u\le -a\), the square root is strictly less than \(|u|\) except at the endpoint, where the sum is \(-a\), still nonzero. For \(u\ge a\), the sum is positive.

On \(-a<u<a\), the bounded difference form has antiderivative

Here \(\arcsin\) denotes the inverse sine with values in \([-\pi/2,\pi/2]\). Its argument \(u/a\) is in \((-1,1)\) on the open interval of validity.

Why the Formulas Work

The first formula follows from integration by parts and the reciprocal-square-root logarithmic form. Write \(R=\sqrt{u^2-a^2}\). On either open interval where \(|u|>a\), differentiation gives \(R'=u/R\). Let \(I=\int R\,du\). Integration by parts with the factors \(u\) and \(R\) gives

Since \(u^2=R^2+a^2\), the last integral splits into \(\int R\,du+a^2\int du/R\). Also,

The last equality follows because \(R+u=u+R\), so \((1+u/R)/(u+R)=(R+u)/(R(u+R))=1/R\). Therefore \(I=uR-I-a^2\ln|u+R|\), and rearranging yields the stated one-half formula.

For the bounded form, write \(S=\sqrt{a^2-u^2}\), so \(S'=-u/S\) on \((-a,a)\). If \(J=\int S\,du\), integration by parts gives

because \(u^2=a^2-S^2\). Since

we obtain \(2J=uS+a^2\arcsin(u/a)\), which proves the second formula.

Worked Example: A Linear Change of Variable in an Unbounded Form

Worked Example: A Bounded Difference Form

Using a Completed Square

A quadratic radical often does not initially display either difference form. If its completed square is \(\alpha(x-h)^2-\beta\), with \(\alpha>0\) and \(\beta>0\), set \(u=\sqrt{\alpha}(x-h)\) and \(a=\sqrt{\beta}\). Then the radicand becomes \(u^2-a^2\), and \(dx=du/\sqrt{\alpha}\). For \(\beta-\alpha(x-h)^2\), the same substitution gives \(a^2-u^2\). The domain must still be stated in the original variable or translated carefully from \(|u|\ge a\) or \(|u|\le a\).

A Common Factor-of-Two Mistake

A frequent error is to use the bracket from the unbounded-form formula without its factor of one-half. For example, with \(R=\sqrt{u^2-4}\), it is incorrect to claim that

is an antiderivative of \(R\). In fact, the derivation shows \(H'(u)=2R\). Therefore the correct antiderivative is \(\frac12H(u)\), not \(H(u)\). A factor outside the bracket from a substitution must then be applied separately; it does not remove the need to verify the bracket’s derivative.

Choosing and Checking the Formula

Before integrating, identify the order of the squared terms. Use the logarithmic formula only for \(u^2-a^2\), where \(|u|\ge a\); use the inverse-sine formula for \(a^2-u^2\), where \(|u|\le a\). Then complete the square, transform the differential, substitute into the appropriate formula, and differentiate the result to check both the sign and the scale factor.

These checks are especially important when \(u\) is a scaled or shifted expression in \(x\). The formula is written in \(u\), but the requested antiderivative is with respect to \(x\). A correct formula in \(u\) can still produce an incorrect answer if the factor \(du/dx\) is omitted.