Calculus I › Integrals › full formula sheet
The secant-squared integral
The derivative of tan x, run in reverse — and its near-twin sec x·tan x is a different rule entirely.
Notation on this page: sec x = 1/cos x, trig functions take radian arguments, and C is the constant of integration.
Before this lesson: Derivative of tan x
Where it comes from
The problem is the reverse of d/dx [tan x] = sec²x — one of the six trig derivatives. Since differentiating tangent produces sec²x, integrating sec²x must produce tangent.
The tempting near-miss:
Before reading on: d/dx [tan x] = sec2x is a derivative fact you already memorized. Read it backwards — what does it immediately tell you about ∫ sec2x dx? Commit before the reveal.
Where does it come from? From a different derivative fact: d/dx [sec x] = sec x·tan x. Students file “sec²” and “sec·tan” in the same mental drawer, and the wrong one slides out. But differentiating the guess does not give sec²x back — and the quotient-rule computation below shows exactly what differentiating tan x does give.
Intuition for the shape: tan x has vertical asymptotes at π/2 + kπ, blowing up to ±∞. Its derivative must blow up there too — and sec²x = 1/cos²x does exactly that, since cos x = 0 at those points. The integrand and the antiderivative share the same asymptotes, which is a strong hint they belong together.
Derivation
We verify F(x) = tan x by differentiating it with the quotient rule. The Pythagorean identity does the final simplification.
Why +C is the whole story: any two antiderivatives of sec²x differ by a constant (zero derivative ⇒ constant), so tan x + C captures every one of them — on each interval between asymptotes.
How to use it
Before reading on: Example 4 below looks almost identical to Example 1: sec x·tan x versus sec2x. Before you read it: whose derivative produces a sec x·tan x factor? (Hint: it is not tangent.)
The procedure, every time:
- Confirm the integrand is sec²x — secant squared, argument exactly x. Not sec x·tan x (that is a different rule), not sec x alone (that is a harder integral).
- Write tan x + C.
- Constants factor out: ∫ 3 sec²x dx = 3 tan x + C.
- Check by differentiating: d/dx [tan x] should give sec²x back.
The near-twin
Memorize the pair together, because exams love to swap them: ∫ sec²x dx = tan x + C but ∫ sec x·tan x dx = sec x + C. The antiderivative is always the “shorter” function: sec² → tan, sec·tan → sec.
The chain-rule fingerprint
∫ sec²(2x) dx = tan(2x)/2 + C — differentiating tan(2x) gives 2 sec²(2x), so divide by 2.
Worked examples
Four problems, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — the basic move: ∫ sec²x dx
- Recognize the pattern. Integrand is sec²x — secant squared, argument exactly x.
- Write the answer: tan x + C. (Why? d/dx [tan x] = sec²x — the derivative fact read backwards.)
- Check: quotient rule on sin x/cos x gives sec²x. Matches ✓
Your turn: Compute ∫ 2 sec2x dx.
Answer: 2 tan x + C
Factor the 2: 2·∫ sec2x dx = 2 tan x + C. Check: d/dx [2 tan x] = 2 sec2x. ✓
Example 2 — definite: ∫0π/4 sec²x dx
- Antiderivative: tan x.
- Evaluate: tan(π/4) − tan 0 = 1 − 0 = 1.
- Sanity check: on [0, π/4], sec²x runs from 1 to 2, so the area should be between 1·(π/4) ≈ 0.79 and 2·(π/4) ≈ 1.57. The answer 1 sits inside. ✓
Your turn: Compute ∫0π/3 sec2x dx.
Answer: √3 ≈ 1.732
Antiderivative tan x: tan(π/3) − tan 0 = √3 − 0 = √3. Sanity: on [0, π/3], sec2x runs 1→4, so the area sits between 1·(π/3) ≈ 1.05 and 4·(π/3) ≈ 4.19. 1.732 fits. ✓
Example 3 — term by term: ∫ (sec²x + x) dx
- Split: ∫sec²x dx + ∫x dx.
- First piece: tan x. Second piece (power rule): x²/2.
- Combine: tan x + x²/2 + C.
- Check: d/dx gives sec²x + x. Matches ✓
Your turn: Compute ∫ (sec2x + x2) dx.
Answer: tan x + x3/3 + C
Split: sec2x integrates to tan x; power rule on x2 gives x3/3. One +C. Check: d/dx gives sec2x + x2. ✓
Example 4 — the near-twin: ∫ sec x·tan x dx
- Notice this is NOT sec²x. The integrand is the product sec x·tan x — the derivative of sec x, not of tan x.
- Verify the guess sec x: d/dx [sec x] = sec x·tan x. Matches ✓
- Answer: sec x + C.
Your turn: Compute ∫ 3 sec x·tan x dx.
Answer: 3 sec x + C
This is the near-twin, not sec2x: d/dx [sec x] = sec x·tan x, so 3·∫ sec x·tan x dx = 3 sec x + C. Check: d/dx [3 sec x] = 3 sec x·tan x. ✓
Memorization tips
- Say it aloud: “sec-squared to tan.” Three syllables, in order.
- Learn the pair: sec² → tan, but sec·tan → sec. The antiderivative is always the “shorter” function — quiz yourself by covering one column.
- Anchor: ∫0π/4 sec²x dx = 1. One clean definite integral.
- The asymptote test: tan x blows up at π/2, and so does sec²x. If your antiderivative does not share the integrand’s blow-ups, something is off.
- Read twice: before writing, check — squared, or product? That one glance defeats the near-twin trap.
Final challenge
Five mixed questions — the near-twin, a √3 answer, and the derivative direction. Score 5/5 and sec²x is yours.
← Back to the Calculus I formula sheet
How to learn a formula here
- Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
- Work the examples with the answers covered, then uncover one step at a time and compare.
- Finish with the final challenge — five mixed questions including the classic traps.
- Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.
Frequently asked questions
What is the integral of sec²x?
∫ sec²x dx = tan x + C, because d/dx [tan x] = sec²x. It is the derivative of tangent, run in reverse.
Why is the antiderivative tan x and not sec x·tan x?
Because d/dx [sec x] = sec x·tan x — that’s the derivative of secant, a different fact. Differentiating tan x gives sec²x, so tan x is the antiderivative.
What is ∫0π/4 sec²x dx?
[tan x]0π/4 = tan(π/4) − tan 0 = 1 − 0 = 1.
What is ∫ sec x·tan x dx?
sec x + C — a close neighbor, not the same rule. d/dx [sec x] = sec x·tan x, so secant (not tangent) is its antiderivative.
How do I remember ∫ sec²x dx = tan x?
Pair it with the derivative you already know: d/dx [tan x] = sec²x. The integral is just that fact read backwards.
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