Recognizing the Pattern
A trigonometric function inside another expression can make an integral look more complicated than it is. When the derivative of the inner function is also present, the reverse chain rule can turn the integral into a familiar basic antiderivative.
For tangent and secant, the most useful derivative identities are
These identities suggest two direct patterns: a secant-squared factor pairs with the derivative of the inner function to produce a tangent, while a secant-times-tangent factor pairs with it to produce a secant. A third pattern comes from the derivative of the logarithm of the absolute value of cosine.
The Three Basic Composite Patterns
Let \(g\) be differentiable on an interval, and suppose the expressions below are defined there. The following formulas apply whenever the displayed factor \(g'(x)\) is present:
The third formula is also commonly written as \(\ln|\sec(g(x))|+C\), since \(|\sec u|=1/|\cos u|\) wherever cosine is nonzero. The absolute value is important: the logarithm in a real-valued antiderivative must have a positive argument.
Each formula is valid on an interval where its integrand and proposed antiderivative are defined. In particular, \(\tan(g(x))\) and \(\sec(g(x))\) require \(\cos(g(x))\ne 0\). If the expression is undefined at some points, apply the formula separately on intervals that do not cross those points.
Why the Formulas Work
The formulas follow by differentiating the proposed antiderivatives. This gives a direct justification and shows exactly why the inner derivative must appear.
By the chain rule, the derivative of \(\tan(g(x))\) is \(\sec^2(g(x))g'(x)\). This is precisely the first integrand.
The derivative of \(\sec(g(x))\) is \(\sec(g(x))\tan(g(x))g'(x)\), which gives the second pattern.
Where \(\cos(g(x))\ne 0\), the derivative of \(-\ln|\cos(g(x))|\) is \(\tan(g(x))g'(x)\), giving the third pattern.
For the logarithmic step in detail, the derivative of \(\ln|v(x)|\), where \(v(x)\ne 0\), is \(v'(x)/v(x)\). Set \(v(x)=\cos(g(x))\). Then
This calculation also explains the domain restriction: the logarithm is defined only when \(\cos(g(x))\ne 0\), exactly the condition needed for the tangent expression to be defined.
Matching Constant Factors
The derivative factor does not need to appear with coefficient exactly one. If \(g'(x)\) is a constant multiple of the factor in the integral, adjust the answer by the reciprocal of that constant. More generally, if the integrand contains \(A g'(x)\) times a trigonometric pattern, then the corresponding antiderivative is multiplied by \(A\).
For example, if the coefficient of a secant-squared pattern is \(k\) and \(g'(x)=m\), with \(m\ne 0\), then the integrand \(k\sec^2(g(x))\) can be written as \((k/m)\sec^2(g(x))g'(x)\). The resulting antiderivative is \((k/m)\tan(g(x))+C\). The same constant adjustment works for the secant-times-tangent and tangent patterns.
Worked Example: A Secant-Squared Composition
Evaluate \(\int 6\sec^2(3x-1)\,dx\). The inner function is \(g(x)=3x-1\), so \(g'(x)=3\). The integrand coefficient \(6\) is twice this derivative:
Apply the secant-squared pattern, keeping the factor \(2\):
This formula holds on any interval where \(\cos(3x-1)\ne 0\). Check by differentiating:
The derivative reproduces the original integrand.
Worked Example: A Secant-Tangent Composition
Evaluate \(\int 4\sec(2x+\pi/6)\tan(2x+\pi/6)\,dx\). Let \(g(x)=2x+\pi/6\), so \(g'(x)=2\). Rewrite the coefficient using this derivative:
The secant-times-tangent pattern therefore gives
The expression is defined on intervals where \(\cos(2x+\pi/6)\ne 0\). To check, differentiate the answer:
This is the original integrand.
Worked Example: A Tangent Composition
Evaluate \(\int \frac{3}{2}\tan(3x-2)\,dx\). The inner function is \(g(x)=3x-2\), and \(g'(x)=3\). Thus \(\frac{3}{2}\tan(g(x))\) is one half of \(\tan(g(x))g'(x)\):
Using the tangent pattern gives
This antiderivative is valid on intervals where \(\cos(3x-2)\ne 0\). Its derivative is
The result matches the integrand, including its coefficient.
Combining Patterns and Respecting Domains
An integrand can contain more than one recognizable term. If each term matches a standard pattern, integrate the terms separately and keep track of where all of them are defined. For a sum of terms, the domain of the combined formula must avoid every point where any term is undefined.
Worked Example: Two Patterns in One Integral
Evaluate \(\int [4\sec^2(2x)+3\sec(3x)\tan(3x)]\,dx\). The first term has inner function \(2x\), whose derivative is \(2\), so its coefficient \(4\) gives a factor of \(2\) in the answer. The second has inner function \(3x\), whose derivative is \(3\), so its coefficient already matches:
Adding the antiderivatives gives
This expression is valid on intervals where both \(\cos(2x)\ne 0\) and \(\cos(3x)\ne 0\). Differentiate to check both terms:
The derivative is exactly the original integrand.
A Common Mistake: Losing the Inner Derivative
A frequent error is to recognize the outer function but forget that differentiating a composition produces an inner derivative factor. For instance, the calculation
is incorrect. Differentiating the proposed answer gives
which is three times the integrand. The correction is to compensate for the inner derivative:
because differentiating this corrected expression gives \(\frac{1}{3}\cdot 3\sec^2(3x-1)=\sec^2(3x-1)\). Always check the proposed result by differentiation; this reveals a missing factor immediately.
When These Patterns Apply
These formulas apply directly when the integrand contains the derivative of its inner function, up to a constant factor. For example, \(\sec^2(g(x))g'(x)\) fits the first pattern. If the inner derivative is absent and is not a constant multiple of the remaining factors, these formulas alone do not evaluate the integral.
Do not ignore domain restrictions when writing an antiderivative. Tangent and secant are undefined where cosine is zero, so a formula involving them describes antiderivatives on intervals that avoid those points. Likewise, the logarithmic tangent antiderivative requires \(\cos(g(x))\ne 0\). Constants of integration may differ from one such interval to another.
The practical value of recognizing these patterns is that a complicated-looking composition can often be integrated by matching only two features: the outer trigonometric pattern and the derivative of the inner function. A final differentiation check confirms both the functional form and every constant factor.
Check Your Understanding
For each integral, identify the inner function, account for its derivative, and state any relevant domain restriction.
- Evaluate \(\int 8\sec^2(4x+1)\,dx\).
- Evaluate \(\int 5\sec(5x)\tan(5x)\,dx\).
- Evaluate \(\int 2\tan(2x+\pi/3)\,dx\).
- On what points is \(\tan(7x-1)\) undefined?
- Why is \(\tan(3x)+C\) not an antiderivative of \(\sec^2(3x)\)?
