Recognizing the Derivative Patterns
Some trigonometric integrals can be found by recognizing that the integrand is already the derivative of a familiar function. In this tutorial, the two patterns are \(\sec x\tan x\) and \(\csc x\cot x\). The first is the derivative of secant; the second is the negative of the derivative of cosecant.
The sign in the cosecant rule is important. Secant and cosecant look similar, but their derivative rules do not give the same sign. A reliable way to remember the antiderivatives is to write the derivative rule first, then reverse it.
Why These Formulas Work
Recall that an antiderivative of a function \(f\) on an interval \(I\) is a function \(F\) defined on \(I\) such that \(F'(x)=f(x)\) for every \(x\in I\). The derivative rules for secant and cosecant show why the two formulas above are valid.
Where \(\cos x\ne 0\), secant is the reciprocal of cosine. Using the derivative of a reciprocal and \((\cos x)'=-\sin x\), we obtain
Where \(\sin x\ne 0\), cosecant is the reciprocal of sine. Since \((\sin x)'=\cos x\), its derivative is
To integrate one of these products, we look for a function whose derivative is the integrand. The first calculation directly identifies \(\sec x\) as an antiderivative of \(\sec x\tan x\). The second identifies \(-\csc x\) as an antiderivative of \(\csc x\cot x\), because
This is the central reasoning: integration reverses differentiation when we already know a function with the required derivative. The constant \(C\) is included because the derivative of any constant is zero, so adding a constant does not change the derivative.
The Antiderivative Formulas
The formulas apply only where the trigonometric functions in the integrand are defined. Secant and tangent require \(\cos x\ne 0\), while cosecant and cotangent require \(\sin x\ne 0\). Therefore, the secant-tangent formula applies on any interval that contains no point \(x=\frac{\pi}{2}+k\pi\), and the cosecant-cotangent formula applies on any interval that contains no point \(x=k\pi\), where \(k\) is an integer.
On such an interval, the antiderivatives can be written in integral notation as follows:
Each formula describes a family of antiderivatives on the interval under consideration. The constant may differ if a problem considers a different interval. The formulas cannot be used across a point where the integrand is undefined, because an antiderivative on an interval must be defined throughout that interval.
When an argument is a multiple of \(x\), the chain rule introduces another factor. For example, differentiating \(\sec(3x)\) gives \(3\sec(3x)\tan(3x)\), not just \(\sec(3x)\tan(3x)\). Consequently, a matching integral must account for the factor of \(3\).
Worked Examples
Worked Example: Integrating Secant Times Tangent
Find \(\int \sec x\tan x\,dx\).
Step 1: Match the integrand to a derivative. The derivative rule is \(\frac{d}{dx}(\sec x)=\sec x\tan x\). Therefore, secant is an antiderivative of the given integrand.
Step 2: Include the arbitrary constant.
Check: Differentiate the proposed antiderivative:
The derivative matches the integrand, so the result is correct wherever \(\cos x\ne 0\).
Worked Example: Integrating Cosecant Times Cotangent
Find \(\int \csc x\cot x\,dx\).
Step 1: Write the derivative rule with its sign. We have \(\frac{d}{dx}(\csc x)=-\csc x\cot x\). The integrand is the negative of this derivative.
Step 2: Reverse the sign. The derivative of \(-\csc x\) is \(\csc x\cot x\), so
Check:
The sign is therefore correct. This formula applies where \(\sin x\ne 0\).
Worked Example: Accounting for a Constant Inside the Argument
Find \(\int 6\sec(2x)\tan(2x)\,dx\).
Step 1: Differentiate a candidate with the composed argument. By the chain rule,
Step 2: Adjust the coefficient. The derivative above has coefficient \(2\), while the integrand has coefficient \(6\). Multiplying the candidate by \(3\) makes its derivative coefficient \(6\):
Therefore,
Check: Differentiating \(3\sec(2x)+C\) gives \(3\cdot 2\sec(2x)\tan(2x)+0=6\sec(2x)\tan(2x)\), exactly the original integrand. The formula is valid on any interval where \(\cos(2x)\ne 0\).
Worked Example: Accounting for a Constant in the Cosecant Pattern
Find \(\int 8\csc(4x)\cot(4x)\,dx\).
Step 1: Use the chain rule for cosecant.
Step 2: Choose a coefficient that produces \(8\). Since the derivative of \(\csc(4x)\) has coefficient \(-4\), multiplying by \(-2\) gives coefficient \(8\):
Thus,
Check: The differentiation above returns the original integrand. The formula applies where \(\sin(4x)\ne 0\).
Why the Constant Is the Only Difference
Finding one antiderivative gives a whole family by adding \(C\). On an interval, any two antiderivatives of the same function differ by a constant. The argument uses the Mean Value Theorem.
This explains why the notation includes \(+C\): it records all functions with the required derivative on the interval, not only one particular antiderivative.
A Common Sign Mistake
A frequent error is to write \(\int \csc x\cot x\,dx=\csc x+C\), treating the cosecant pattern like the secant pattern. Differentiation exposes the problem:
which is the negative of the integrand, not the integrand itself. The correction is to place a negative sign in front of cosecant:
Checking by differentiation is especially useful for this pair of formulas because the sign is easy to overlook. It also catches missing chain-rule factors when the argument is, for example, \(2x\) or \(4x\).
Summary
The secant-tangent and cosecant-cotangent products are direct derivative patterns. Their antiderivatives follow by reversing the corresponding derivative rules:
When the argument is a multiple of \(x\), include the chain-rule factor in the calculation. In every case, check the proposed antiderivative by differentiating it, and apply the formula only on an interval where the trigonometric expressions are defined.
Check Your Understanding
Try each question, then check your reasoning by differentiating any proposed antiderivative.
- What is an antiderivative of \(\sec x\tan x\), and what is its domain restriction?
- Why does an antiderivative of \(\csc x\cot x\) require a negative sign?
- Find \(\int 10\sec(5x)\tan(5x)\,dx\).
- Find \(\int 3\csc(2x)\cot(2x)\,dx\).
- Why do two antiderivatives of the same function on an interval differ by a constant?
