Checking Sine-Cosine Antiderivatives by Differentiation

Why Check by Differentiation?

An antiderivative can look plausible and still contain a small error. A missing coefficient, an incorrect exponent, or a forgotten factor from a multiple-angle argument can change the derivative—and therefore change the function being integrated.

Differentiation gives a direct test. If a proposed function differentiates to the stated integrand throughout the interval under consideration, it is an antiderivative there. For products of sine and cosine powers, the product rule and chain rule make this test systematic.

The task here is not to develop a method for finding every such antiderivative. It is to check a proposed answer carefully: differentiate it, simplify the result, and compare it with the integrand.

The Derivative Pattern for Sine and Cosine Powers

Let \(\theta=kx+\phi\), where \(k\) and \(\phi\) are constants. For nonnegative integer powers \(m\) and \(n\), the product and chain rules give the following derivative when \(m,n>0\):

To see the two contributions, differentiate the sine power while leaving the cosine power unchanged, then differentiate the cosine power while leaving the sine power unchanged:

The factor \(k\) comes from \(\frac{d\theta}{dx}=k\). The minus sign in the second term comes from the derivative of cosine. If one exponent is zero, differentiate the nonconstant factor directly; the displayed formula can also be interpreted by omitting the term multiplied by that zero exponent.

A proposed answer may be a sum of several terms, or it may involve a multiple-angle function. Differentiate every term, preserve all coefficients, and simplify only after the differentiation is complete. Then compare the simplified result with the original integrand.

Worked Example: Checking a Polynomial in Cosine

Follow the Angle Through the Chain Rule

When a sine or cosine contains \(kx+\phi\), differentiating the inside contributes a factor of \(k\). That factor must be matched when checking the coefficient in front of the proposed antiderivative. The next example isolates this issue.

Check Multiple-Angle Terms Carefully

Multiple-angle identities can produce proposed antiderivatives containing terms such as \(\sin(4x)\). Their coefficients must account for both the coefficient of the sine term and the derivative of its angle. Differentiate the proposed expression directly rather than guessing whether a factor has been included.

A Reliable Checking Procedure

A derivative check is most useful when it is done in a fixed order. The following steps help separate the chain-rule calculation from the algebraic comparison.

Why the Derivative Test Is Decisive

The definition of antiderivative makes the first part of the test immediate: if \(F'(x)=f(x)\) at every point of an interval, then \(F\) is an antiderivative of \(f\) on that interval. The test also explains why two correct answers can differ by a constant.

Suppose \(F\) and \(G\) are differentiable on an interval \(I\) and both have derivative \(f\). Define \(H=F-G\). Then, for every \(x\in I\),

For any two points \(a,b\in I\) with \(a<b\), the Mean Value Theorem applies to \(H\) on \([a,b]\): \(H\) is continuous on that closed interval and differentiable on its interior. It gives a point \(c\in(a,b)\) such that

Thus \(H(b)=H(a)\). Since the points were arbitrary, \(H\) is constant on \(I\), so \(F\) and \(G\) differ by a constant there. This is why checking the derivative, rather than comparing two formulas term by term, is the decisive test.

Common Pitfall: Omitting the Multiple-Angle Factor

Consider \(\sin^2(3x)\cos(3x)\). A tempting proposed antiderivative is \(\frac{\sin^3(3x)}{3}\), but this omits the effect of differentiating \(3x\). Differentiate the proposal to see the exact error:

The derivative is three times the desired integrand, not the integrand itself. To correct the coefficient, use \(\frac{\sin^3(3x)}{9}\). Checking this revision:

The revised expression passes the derivative test. The correction comes from accounting for both the derivative of the cube and the factor \(3\) from the inner angle.

What the Check Can Tell You

If the derivative has the right powers and angles but the wrong numerical multiple, inspect the proposed coefficient and any chain-rule factor. If a power is wrong, check how differentiating a power changes its exponent: differentiating \(\sin^m x\) lowers the sine exponent by one and introduces a cosine, while differentiating \(\cos^n x\) lowers the cosine exponent by one, introduces a sine, and contributes a minus sign.

Sometimes the differentiated expression does not look like the integrand until it is factored or simplified with a trigonometric identity. That is not a reason to declare the candidate incorrect. Compare equivalent expressions, but do not use an identity to conceal a missing factor or sign. A successful check ends with an explicit equality between the derivative and the stated integrand.