Integrating Sine-Cosine Products at Different Frequencies

Why Convert a Product into a Sum?

Products such as \(\sin(5x)\sin(2x)\) and \(\sin(3x)\cos(2x)\) are not in the basic form of a single sine or cosine function. Product-to-sum identities rewrite each product as a sum of simpler trigonometric terms. Each term can then be integrated directly, provided its frequency is not zero.

The key detail is that integrating a trigonometric function with frequency \(k\) introduces a factor of \(1/k\). In products with distinct positive frequencies \(a\) and \(b\), the resulting frequencies are \(a+b\) and \(a-b\). The difference can be negative, but it is nonzero when \(a\ne b\); keeping its sign in the formula avoids sign errors.

Identities and the Frequency Condition

The product-to-sum identities express products of two trigonometric functions as sums. They follow from the angle-sum and angle-difference identities; here they are stated in the forms needed for integration.

For any nonzero real number \(k\), the chain rule gives the basic antiderivatives

The denominator is the signed frequency \(k\), not its absolute value. If \(a\ne b\), both \(a+b\) and \(a-b\) are nonzero, so these rules apply to every term in the identities. If \(a=b\), the difference frequency is zero, and the rule for a nonzero frequency cannot be used on that term.

Antiderivative Formulas for Distinct Frequencies

Integrating the three product-to-sum identities term by term gives the formulas below. Their signed denominators make them valid whether \(a>b\) or \(b>a\).

Why the Formulas Work

A reliable justification is to differentiate the proposed antiderivative and recover the original product. This checks both the signs and the frequency factors. For example, let

By the chain rule, the derivative of \(\cos(kx)\) is \(-k\sin(kx)\). Therefore

where the last equality is the product-to-sum identity for sine times cosine. Thus \(F\) is an antiderivative of the original product. The other two formulas are checked in the same way: differentiating \(\sin(kx)/k\) gives \(\cos(kx)\), and the corresponding product-to-sum identity recovers the product.

This argument also explains the denominators. When a term has frequency \(k\), differentiating its sine or cosine produces a factor of \(k\). Dividing by \(k\) in the antiderivative cancels that factor. A zero frequency is different: \(\cos(0x)=1\), whose antiderivative is \(x\), not \(\sin(0x)/0\).

Worked Examples

When the Frequencies Match

The condition \(a\ne b\) matters. If \(a=b\), the difference-frequency term becomes a constant, because \(\cos((a-a)x)=1\). Its integral is \(x\). The product-to-sum identities therefore give the following special cases for positive \(a\):

For instance, the sine-sine identity becomes \(\sin^2(ax)=\tfrac12[1-\cos(2ax)]\). Integrating the constant term \(\tfrac12\) produces \(x/2\), while integrating the other term produces \(-\sin(2ax)/(4a)\). The zero-frequency case is handled directly; it is not obtained by substituting \(a=b\) into a formula with denominator \(a-b\).

Common Pitfalls

A frequent mistake is reversing the difference terms in the sine-sine identity. For example, the following calculation is wrong:

The correct order is the cosine of the difference minus the cosine of the sum:

The sign error is easy to detect by setting \(x=0\). The original product is zero, and the correct right-hand side is \(\tfrac12(1-1)=0\). The incorrect right-hand side would be \(\tfrac12(1-1)=0\) too, so this particular check does not distinguish the two orders; instead, differentiate the proposed antiderivative or test a value such as \(x=\pi/7\). At that value, \(\cos(7x)=\cos(\pi)=-1\), while \(\cos(3x)=\cos(3\pi/7)\), and reversing the terms changes the result. The identity as stated follows from the product-to-sum formula and should be retained in that order.

Another common error is omitting a frequency factor during integration. For example, \(\int \cos(5x)\,dx\) is not \(\sin(5x)\); differentiating \(\sin(5x)\) gives \(5\cos(5x)\). The correct antiderivative is \(\sin(5x)/5+C\). Finally, do not replace \(a-b\) by \(|a-b|\) in a formula without also tracking how the trigonometric term changes. Using the signed frequency as written is the safest approach.

A Reliable Procedure

For a product with distinct positive integer frequencies, keep the calculation organized in three stages:

Converting the product before integrating makes the calculation transparent: every term is a basic sine or cosine, every denominator comes from its frequency, and differentiation provides a direct check. The main decision is not a complicated integration technique; it is correctly identifying the sum and difference frequencies and respecting the zero-frequency exception.