Why the Choice of Factor Matters
A mixed tangent-secant integral contains both powers of \(\tan x\) and powers of \(\sec x\). The main task is to arrange the factors so that one part is the derivative of a useful substitution, while the remaining part can be expressed in terms of that substitution.
Two derivative identities guide the choices in this tutorial:
The identity \(\sec^2 x=1+\tan^2 x\) connects the two functions. Depending on the integrand, it can turn the leftover factors into a polynomial in either \(\tan x\) or \(\sec x\). The substitution should be chosen to use the factors already present, not imposed before examining the integrand.
A Substitution Principle for These Integrals
If \(u=g(x)\), then \(du=g'(x)\,dx\). The substitution is useful when the integrand can be written as a function of \(g(x)\) multiplied by \(g'(x)\). The reason is the chain rule: if \(F'(u)=f(u)\), then
Thus, once the integrand has the form \(f(g(x))g'(x)\), integrating \(f(u)\) with respect to \(u\) produces an antiderivative in \(x\). In the examples, the algebraic identity is what makes the leftover factors a function of the substituted variable.
Worked Example: \(\int \tan^2 x\sec^4 x\,dx\)
Worked Example: Use \(u=\tan x\)
Consider
A factor of \(\sec^2 x\,dx\) is present, and it is the differential of \(\tan x\). Separate that factor from the other powers:
Set \(u=\tan x\), so \(du=\sec^2 x\,dx\). In the remaining factor, use \(\sec^2 x=1+\tan^2 x\):
The integral is now a polynomial integral:
Substitute \(u=\tan x\) back:
Check by differentiating. Since \((\tan x)'=\sec^2 x\),
The last equality uses \(1+\tan^2 x=\sec^2 x\), so the derivative matches the integrand.
When the Secant Substitution Fits
The derivative of \(\sec x\) contains the paired factor \(\sec x\tan x\). If an integrand has one factor of each, setting \(u=\sec x\) may leave the other factors expressible using \(\tan^2 x=\sec^2 x-1\). This is especially helpful when an odd power of tangent provides a factor \(\tan x\), while the remaining tangent powers can be grouped in pairs.
Worked Example: \(\int \tan^3 x\sec^3 x\,dx\)
Worked Example: Use \(u=\sec x\)
Evaluate
Reserve a factor \(\tan x\sec x\,dx\), which is \(d(\sec x)\). The remaining factor is \(\tan^2 x\sec^2 x\):
Set \(u=\sec x\), so \(du=\sec x\tan x\,dx\). Rewrite the remaining tangent square using \(\tan^2 x=\sec^2 x-1\):
Therefore,
Returning to \(x\) gives
Differentiate to check the result:
Here the third line uses \(\sec^2 x-1=\tan^2 x\), confirming that the derivative is the original integrand.
Worked Example: \(\int \tan^4 x\sec^2 x\,dx\)
Worked Example: Use \(u=\tan x\) directly
Consider
This integrand already contains the complete differential for \(u=\tan x\). Set \(u=\tan x\), so \(du=\sec^2 x\,dx\). No identity is needed for the remaining tangent power:
Differentiate the proposed antiderivative:
The result checks exactly. This example illustrates that the simplest substitution is often the one whose derivative factor is already present as a complete factor in the integrand.
A Practical Way to Organize the Algebra
These examples differ in which derivative factor is set aside, but their reasoning follows the same sequence. First identify a derivative factor; next rewrite any leftover powers in the chosen variable; then integrate; finally differentiate the answer to check it.
Look for \(\sec^2 x\,dx\), associated with \(d(\tan x)\), or \(\sec x\tan x\,dx\), associated with \(d(\sec x)\).
Separate it explicitly so it is clear what becomes \(du\).
Use \(\sec^2 x=1+\tan^2 x\) for a tangent substitution, or \(\tan^2 x=\sec^2 x-1\) for a secant substitution.
Integrate the resulting expression in \(u\), substitute back, and differentiate the answer.
Common Pitfall: Losing a Leftover Factor
When using \(u=\tan x\), only one factor of \(\sec^2 x\,dx\) becomes \(du\). Any additional secant factors must still be converted into expressions involving \(u\). For the first example, the following calculation is incorrect:
This drops the remaining factor \(\sec^2 x\). The correct separation is
so the leftover factor becomes \(u^2(1+u^2)\), not merely \(u^2\). The incorrect proposed answer also fails its derivative check:
in general. Differentiating is a direct way to reveal that a factor was lost.
Why These Substitutions Are Useful
Mixed tangent-secant expressions can look complicated because the same trigonometric functions appear in several powers. A suitable substitution reduces the integral to an ordinary polynomial integral: the derivative factor accounts for \(dx\), and a Pythagorean identity converts what remains into the chosen variable. The method is reliable when both pieces are handled—the derivative factor and every leftover factor.
The three worked integrals show the main distinctions. In \(\int \tan^2 x\sec^4 x\,dx\), the tangent substitution leaves an extra secant square to rewrite. In \(\int \tan^3 x\sec^3 x\,dx\), the secant substitution uses the paired tangent-secant factor. In \(\int \tan^4 x\sec^2 x\,dx\), the tangent substitution is already complete. In every case, differentiating the final expression confirms the computation.
Check Your Understanding
For each question, identify the useful substitution or the identity needed before carrying out the integration.
- For \(\int \tan^2 x\sec^4 x\,dx\), which factor becomes \(du\) under \(u=\tan x\), and how should the leftover secant square be rewritten?
- Why is \(u=\sec x\) a useful substitution for \(\int \tan^3 x\sec^3 x\,dx\)? State the differential factor it uses.
- Evaluate \(\int \tan^4 x\sec^2 x\,dx\) and check the result by differentiation.
- What factor is lost in the incorrect calculation \(\int \tan^2 x\sec^4 x\,dx=\int u^2\,du\) with \(u=\tan x\)?
- State the identity that converts leftover tangent squares into powers of \(\sec x\).
