Calculus I › Integrals › full formula sheet
The cosine integral
No minus sign this time — and the #1 trap is importing one from the derivative.
Notation on this page: trig functions take radian arguments, and C is the constant of integration.
Before this lesson: Derivative of sin x · Derivative of cos x
Where it comes from
The problem is the reverse of d/dx [sin x] = cos x — the one trig derivative with no minus sign. Running it backwards, the integral of cosine is sine, equally minus-free.
The tempting import:
Before reading on: you just learned ∫ sin x dx = −cos x + C. Which of the two trig integrals do you think carries the minus — and how would you check your guess in five seconds?
Where does this wrong answer come from? From the derivative of cosine: d/dx [cos x] = −sin x. Students see “cosine” and “minus” together so often that the minus leaks into the integral. But differentiation and integration run in opposite directions:
Intuition for the shape: cos x is positive on (−π/2, π/2), so its antiderivative must be increasing there — and sin x indeed climbs from −1 to 1 across that interval. The geometry agrees: no sign flip needed.
Derivation
Guess-and-verify: propose F(x) = sin x and differentiate. This time there is no double negative — the verification is one clean step.
Why +C is the whole story: any two antiderivatives of cos x differ by a constant (zero derivative ⇒ constant), so sin x + C captures every one of them.
How to use it
Before reading on: on [0, π/2], cos x falls from 1 to 0. Before the reveal: is the area under it bigger or smaller than 1? (The reveal says it is exactly 1 — what makes that plausible?)
The procedure, every time:
- Confirm the integrand is cos x alone (times constants). If the argument is 5x or x², the chain rule left a fingerprint — see below.
- Write sin x + C. No minus. Say “cosine to sine” as you write it.
- Constants factor out: ∫ 2 cos x dx = 2 sin x + C.
- Check by differentiating: d/dx [sin x] = cos x. If a minus appears in your check, you imported it — delete it.
The chain-rule fingerprint
∫ cos(5x) dx is not sin(5x) + C: differentiating sin(5x) gives 5 cos(5x) — five times too big. Divide by 5: ∫ cos(5x) dx = sin(5x)/5 + C. This is u-substitution in disguise (u = 5x).
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: ∫ cos x dx
- Recognize the pattern. Integrand is cos x with argument exactly x.
- Write the answer: sin x + C. (Why no minus? d/dx [sin x] = cos x — clean, no sign to fix.)
- Check: d/dx [sin x] = cos x. Matches ✓
Your turn: Compute ∫ 4 cos x dx.
Answer: 4 sin x + C
Factor the 4: 4·∫ cos x dx = 4 sin x + C — no minus. Check: d/dx [4 sin x] = 4 cos x. ✓
Example 2 — definite: ∫0π/2 cos x dx
- Antiderivative: sin x.
- Evaluate: sin(π/2) − sin 0 = 1 − 0 = 1.
- Sanity check: cos x falls from 1 to 0 on [0, π/2], so the area should be between 0 and 1·(π/2) ≈ 1.57, and below the triangle of area π/4 ≈ 0.79? No — cos x bows outward above the triangle, so the area should exceed 0.79. The answer 1 fits. ✓
Your turn: Compute ∫0π cos x dx.
Answer: 0
Antiderivative sin x: sin π − sin 0 = 0 − 0 = 0. Sanity: cos x is positive on [0, π/2) and negative on (π/2, π], symmetric — the lobes cancel exactly. ✓
Example 3 — term by term: ∫ (x + cos x) dx
- Split: ∫x dx + ∫cos x dx.
- First piece (power rule): x²/2. Second piece: sin x (no minus!).
- Combine: x²/2 + sin x + C.
- Check: d/dx gives x + cos x. Matches ✓
Your turn: Compute ∫ (x2 + cos x) dx.
Answer: x3/3 + sin x + C
Split: power rule on x2 gives x3/3; cosine integrates to sin x (no minus). One +C. Check: d/dx gives x2 + cos x. ✓
Example 4 — initial value: find F with F′ = cos x and F(0) = 0
- General antiderivative: F(x) = sin x + C.
- Use the condition: F(0) = sin 0 + C = 0 + C = 0, so C = 0.
- Answer: F(x) = sin x.
- Check: F′ = cos x ✓; F(0) = 0 ✓.
Your turn: Find F with F′ = cos x and F(π) = 2.
Answer: F(x) = sin x + 2
General antiderivative: F(x) = sin x + C. Condition: F(π) = sin π + C = 0 + C = 2, so C = 2. Check: F′ = cos x ✓; F(π) = 2 ✓.
Memorization tips
- Say it aloud: “cosine to sine — clean.” Then the pair: “sine to negative cosine; cosine to sine.”
- The direction test: whenever a minus tempts you, ask “am I differentiating or integrating?” The minus belongs to differentiating cosine.
- The 5-second check: differentiate your answer. If a minus appears that was not in the integrand, you imported it — delete it.
- Anchor: ∫0π/2 cos x dx = 1. One clean definite integral, no signs to fumble.
- Learn the pair, not the singles: sine and cosine integrals are one fact (“the minus belongs to sine’s integral”), not two facts to memorize separately.
Final challenge
Five mixed questions — the minus trap, signed areas, and a motion problem. Score 5/5 and the cosine integral 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 cos x?
∫ cos x dx = sin x + C. No minus sign: differentiating sin x gives exactly cos x.
Why is there no minus sign in ∫ cos x dx?
Because d/dx [sin x] = cos x with no minus involved. The minus sign appears in the derivative of cosine, not in the integral of cosine.
Is ∫ cos x dx = −sin x + C?
No — that confuses the integral with the derivative. d/dx [cos x] = −sin x is differentiation; ∫ cos x dx = sin x + C is integration. They are opposite directions.
What is ∫0π/2 cos x dx?
[sin x]0π/2 = sin(π/2) − sin 0 = 1 − 0 = 1.
How do I remember which trig integral has the minus?
Pair them: “sine to negative cosine; cosine to sine.” The minus belongs to sine’s integral. Cosine’s integral is clean.
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