Integrals of Powers of Secant and Tangent
Textbook Reference
| Primary source | OpenStax Calculus Volume 2, Section 3.2: “Trigonometric Integrals” |
| Direct link | https://openstax.org/books/calculus-volume-2/pages/3-2-trigonometric-integrals |
| Textbook used in class | Stewart, Calculus, Section 7.2: “Trigonometric Integrals” |
Opening Scenario
Integrating powers of $\sin x$ and $\cos x$ depended on the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to convert between them. For $\sec x$ and $\tan x$, the same principle applies but with the identity $\sec^2 x = 1 + \tan^2 x$ and the derivative pair $\dfrac{d}{dx}(\tan x) = \sec^2 x$ and $\dfrac{d}{dx}(\sec x) = \sec x\tan x$.
Quick Reference
Key derivative facts:
- $\dfrac{d}{dx}(\tan x) = \sec^2 x$, so $\displaystyle\int \sec^2 x\,dx = \tan x + C$.
- $\dfrac{d}{dx}(\sec x) = \sec x\tan x$, so $\displaystyle\int \sec x\tan x\,dx = \sec x + C$.
Key identity: $\sec^2 x = 1 + \tan^2 x$.
Standard result: $\displaystyle\int \sec x\,dx = \ln|\sec x + \tan x| + C$.
Strategies for $\displaystyle\int \sec^m x\,\tan^n x\,dx$:
| Case | Strategy |
|---|---|
| $n$ odd | Factor out $\sec x\tan x$; convert remaining even powers of $\tan x$ using $\tan^2 x = \sec^2 x - 1$; let $u = \sec x$ |
| $m$ even | Factor out $\sec^2 x$; convert remaining even powers of $\sec x$ using $\sec^2 x = 1 + \tan^2 x$; let $u = \tan x$ |
| $m$ odd, $n$ even | Use integration by parts or a reduction formula |
Key Concepts
1. The Two Substitution Pairings
Two natural substitutions arise from the derivative facts above:
- $u = \tan x$, $du = \sec^2 x\,dx$: useful when $\sec^2 x$ can be isolated as a factor.
- $u = \sec x$, $du = \sec x\tan x\,dx$: useful when $\sec x\tan x$ can be isolated as a factor.
The strategies are designed to engineer one of these situations.
2. The Odd-$n$ Case
If $\tan x$ appears to an odd power, peel off one $\tan x$ and pair it with one $\sec x$ to form the differential $d(\sec x) = \sec x\tan x\,dx$. Convert the remaining even power of $\tan x$ using $\tan^2 x = \sec^2 x - 1$.
Example. $\displaystyle\int \tan^3 x\,\sec x\,dx$.
Factor: $\displaystyle\int \tan^2 x\,(\sec x\tan x)\,dx = \int(\sec^2 x - 1)\,(\sec x\tan x)\,dx$.
Let $u = \sec x$, $du = \sec x\tan x\,dx$:
$\displaystyle\int(u^2 - 1)\,du = \frac{u^3}{3} - u + C = \frac{\sec^3 x}{3} - \sec x + C$.
3. The Even-$m$ Case
If $\sec x$ appears to an even power, peel off $\sec^2 x$ and convert any remaining even powers of $\sec x$ using $\sec^2 x = 1 + \tan^2 x$.
Example. $\displaystyle\int \sec^4 x\,\tan^3 x\,dx$.
$= \displaystyle\int \sec^2 x\,(1 + \tan^2 x)\,\tan^3 x\,\sec^2 x\,dx$... wait, let us redo cleanly:
Peel off $\sec^2 x$: $\displaystyle\int \sec^2 x\cdot\sec^2 x\cdot\tan^3 x\,dx = \int(1 + \tan^2 x)\tan^3 x\,\sec^2 x\,dx$.
Let $u = \tan x$, $du = \sec^2 x\,dx$:
$\displaystyle\int(1 + u^2)u^3\,du = \int(u^3 + u^5)\,du = \frac{u^4}{4} + \frac{u^6}{6} + C = \frac{\tan^4 x}{4} + \frac{\tan^6 x}{6} + C$.
“The two strategies cover every case of $\int \sec^m x\tan^n x\,dx$.” They do not. When $m$ is odd and $n$ is even (for example, $\int\sec^3 x\,dx$ or $\int\sec x\,dx$), neither strategy applies cleanly. These cases require integration by parts, a reduction formula, or a multiplication-and-division trick. Recognizing this boundary is part of the skill.
Worked Example
Evaluate $\displaystyle\int \sec^3 x\,dx$.
This is the case $m = 3$ (odd), $n = 0$ (even). Neither standard strategy applies. Use integration by parts.
Set $u_1 = \sec x$, $dv = \sec^2 x\,dx$, so $v = \tan x$ and $du_1 = \sec x\tan x\,dx$:
$\displaystyle\int\sec^3 x\,dx = \sec x\tan x - \int\tan^2 x\sec x\,dx$.
$= \sec x\tan x - \int(\sec^2 x - 1)\sec x\,dx$
$= \sec x\tan x - \int\sec^3 x\,dx + \int\sec x\,dx$.
Add $\int\sec^3 x\,dx$ to both sides:
$2\int\sec^3 x\,dx = \sec x\tan x + \ln|\sec x + \tan x| + C_0$.
$$\int\sec^3 x\,dx = \frac{1}{2}\sec x\tan x + \frac{1}{2}\ln|\sec x + \tan x| + C.$$
This result appears repeatedly in trigonometric substitution problems.
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Trying $u = \tan x$ when $n$ is odd | $\int\tan^3 x\sec x\,dx$ with $u = \tan x$ | $du = \sec^2 x\,dx$ is not present; use $u = \sec x$ (odd-$n$ strategy) |
| Leaving $\tan^2 x$ unconverted | After factoring out $\sec x\tan x$, integrating $\int\tan^2 x\cdot du$ | Replace $\tan^2 x = u^2 - 1 = \sec^2 x - 1$ before integrating |
| Applying substitution to the $m$-odd, $n$-even case | Trying to substitute in $\int\sec^3 x\,dx$ | Recognize this requires integration by parts |
Leveled Practice
Level 1 -- Even-$m$ Strategy
Problem 1. Evaluate $\displaystyle\int \tan x\,\sec^2 x\,dx$.
Show answer
$u = \tan x$, $du = \sec^2 x\,dx$.
$\displaystyle\int u\,du = \frac{u^2}{2} + C = \frac{\tan^2 x}{2} + C$.
Level 2 -- Odd-$n$ Strategy
Problem 2. Evaluate $\displaystyle\int \tan^3 x\,\sec^3 x\,dx$.
Show answer
$n = 3$ is odd. Factor $\sec x\tan x$:
$\displaystyle\int\tan^2 x\sec^2 x\cdot(\sec x\tan x)\,dx = \int(\sec^2 x - 1)\sec^2 x\cdot(\sec x\tan x)\,dx$.
$u = \sec x$, $du = \sec x\tan x\,dx$:
$\displaystyle\int(u^2-1)u^2\,du = \int(u^4 - u^2)\,du = \frac{u^5}{5} - \frac{u^3}{3} + C = \frac{\sec^5 x}{5} - \frac{\sec^3 x}{3} + C$.
Mastery Checklist
Mental Model
The $\sec$/$\tan$ pair behaves like the $\sin$/$\cos$ pair. In that system, $\sin^2 x + \cos^2 x = 1$ converts between them and the derivatives of one are (signed) multiples of the other. In this system, $\sec^2 x = 1 + \tan^2 x$ converts between them and $\frac{d}{dx}(\tan x) = \sec^2 x$, $\frac{d}{dx}(\sec x) = \sec x\tan x$. Substitute the right variable to reduce the integral to a polynomial in $u$.
Connections
Looking back
- Trig integrals -- sin/cos (Section 7.2): The parallel strategy using the $\sin^2 + \cos^2 = 1$ identity.
- Integration by parts (Section 7.1): Required for the $m$-odd, $n$-even case.
Looking ahead
- Trigonometric substitution (Section 7.3): The formula $\int\sec^3 x\,dx$ appears frequently as a sub-problem.
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