Frequency and Phase in Polynomial-Trigonometric Tables

Why Frequency and Phase Matter

Tabular integration is especially effective when a polynomial is multiplied by a sine or cosine: successive derivatives of the polynomial eventually become zero, while successive antiderivatives of the trigonometric factor cycle through sine and cosine. If the angle is \(bx+c\), however, each antiderivative also introduces a factor involving \(b\). Keeping track of that factor is essential.

The constant \(c\) shifts the phase of the oscillation. It does not change the frequency factor in the integration, but it must remain inside every sine and cosine that appears in the table. We will write the angle as \(\theta=bx+c\) to make those changes easier to follow.

Set Up the Table

Let \(n\) be a positive integer, \(b\) a nonzero real number, and \(c\) a real number. (If \(n=0\), the polynomial factor is the constant \(1\), and its derivative column is just \(1,\ 0\).) The polynomial side of the table is formed by differentiating \(x^n\) until it reaches zero:

On the trigonometric side, begin with the function being integrated and repeatedly take antiderivatives with respect to \(x\). For example, starting with \(\sin(bx+c)\), the first few entries are

Starting instead with \(\cos(bx+c)\), the sequence is

The entries are antiderivatives, not derivatives. For instance, differentiating \(-\cos(bx+c)/b\) gives \(\sin(bx+c)\), because the chain rule contributes a factor of \(b\), which cancels the denominator.

To read the table, multiply diagonally, starting with the original polynomial and the first antiderivative, then alternate plus and minus signs. Continue until the polynomial derivatives reach zero. Each term uses the next antiderivative in the trigonometric sequence.

The General Formulas

The table can be summarized using phase shifts. In the formulas below, \(k\) counts the number of derivatives taken on the polynomial, and the factorial quotient gives the coefficient of that derivative.

Here \(C\) is an arbitrary constant of integration. The phase-shift notation is a compact way to record the changing sine and cosine signs. It does not change the argument \(\theta=bx+c\) itself.

Why the Table Works

A complete justification comes from differentiating the sum. Let \(g(x)\) be either \(\sin(bx+c)\) or \(\cos(bx+c)\). Define \(W_0(x)=g(x)\), and for each \(j\geq 1\), let \(W_j\) be an antiderivative of \(W_{j-1}\), choosing any constants of integration to be zero. Thus \(W_j'=W_{j-1}\), and \(W_j\) is the \(j\)-th successive antiderivative of \(g\).

For \(u(x)=x^n\), form the finite alternating sum

Differentiate each product using the product rule. The result has two sums:

In the first sum, replace the index \(k+1\) by \(j\). Its terms then cancel the terms in the second sum for \(j=1,\ldots,n\), since their signs are opposite. The term left from the second sum at \(j=0\) is \(uW_0=ug\). The only possible final term from the first sum involves \(u^{(n+1)}\), and it is zero because the \((n+1)\)-st derivative of \(x^n\) vanishes. Consequently, \(F'=ug=x^ng\), proving that the alternating table sum is an antiderivative.

For the sine and cosine cases, the chain rule supplies the frequency factors. In particular, the \(j\)-th successive antiderivative of \(g\) contains \(b^{-j}\). Explicitly, those antiderivatives are

Substituting these expressions and \(u^{(k)}=n!x^{n-k}/(n-k)!\) into the alternating sum gives the two stated formulas. The proof also explains why the method stops cleanly: it is the polynomial derivative, not the trigonometric sequence, that eventually becomes zero.

Worked Examples

A Common Frequency Error

A frequent mistake is to take an antiderivative of a trigonometric function of \(bx+c\) as if the derivative of its angle were \(1\). For example, the first antiderivative of \(\sin(3x+\pi/6)\) is not \(-\cos(3x+\pi/6)\). That expression differentiates to \(3\sin(3x+\pi/6)\), three times the desired function.

For a concrete table, consider \(\int x\sin(3x+\pi/6)\,dx\). Omitting the frequency factors gives the incorrect expression \(-x\cos\theta+\sin\theta\), where \(\theta=3x+\pi/6\). Its derivative is \(-\cos\theta+3x\sin\theta+3\cos\theta\), which does not equal \(x\sin\theta\). Keeping the factors of \(1/3\) gives the correct result:

Differentiating this expression gives \(-\cos\theta/3+x\sin\theta+\cos\theta/3=x\sin\theta\). The check catches the missing-frequency error immediately.

When the Formula Applies

The polynomial must have a finite degree, and the frequency must satisfy \(b\ne0\). If \(b=0\), then the trigonometric factor is constant in \(x\), so the displayed formulas—which divide by powers of \(b\)—do not apply. In that case, integrate the constant multiple of \(x^n\) directly.

The phase \(c\) can be any real number. It is retained in every angle and causes no additional factor because differentiating \(bx+c\) still gives \(b\). The frequency \(b\) may be negative as well as positive: the same formulas work because the signed powers of \(b\) correctly account for the chain rule.

These tables provide an efficient, auditable way to integrate a polynomial times a single sine or cosine. Writing out the derivatives and antiderivatives before forming the diagonal sum helps prevent sign errors; differentiating the final expression confirms both the phase and every frequency factor.