Lesson 08 · Formula fluency · 11 min
Basic antiderivative rules
Every antiderivative rule is a derivative rule read backwards. Master one reverse move — raise the power, divide by the new power — plus a few special characters, and the essentials are yours.
Build the intuition
The reverse power rule
The power rule said: bring down, reduce. Its mirror says: raise the power by one, divide by the new power. Check by differentiating — your answer should hand back exactly what you started with. That check is always available, always decisive.
The n = −1 exception
Raise x⁻¹ by one and you'd divide by zero — the pattern jams. The gap is filled by an old friend: the antiderivative of 1/x is ln|x|. The one power the rule can't touch belongs to the logarithm.
The specials, rewound
eˣ integrates to itself (of course). The trig cycle runs backwards: ∫cos = sin, and ∫sin = −cos. When a sign feels uncertain, don't memorize harder — differentiate your answer and let it confess.
See it move
Antiderivatives are about accumulating a rate back into a total. Watch slices of speed pile up into distance.
A worked example
Integrate a polynomial
Take
Reverse power rule, term by term:
Verify by differentiating: 3x² + 2x − 5. ✓ The check costs five seconds and catches everything.
The antiderivative cards — tap any to open
Accumulating a steady rate k gives steady growth — a line with slope k.
Accumulating a steady rate k gives steady growth — a line with slope k.
Memory hook. Steady rate → straight line. Driving at 60 for x hours covers 60x miles.
On the graph. The area under a flat line of height k grows linearly: width × height = kx.
Watch out. Forgetting +C — and forgetting that ∫5 dx has an x in the answer.
Used for. Total cost from a fixed rate: hours × hourly rate.
Try it: What is ∫ 3 dx?
Accumulating a steadily growing rate gives quadratic growth.
Accumulating a steadily growing rate gives quadratic growth.
Memory hook. Reverse power rule: raise the power (1→2), divide by the new power (÷2).
On the graph. Area under y = x up to x is a triangle: ½ · base · height = x²/2.
Watch out. Forgetting to divide — ∫x dx is not x².
Used for. Distance under constant acceleration: speed grows like t, distance like t²/2.
Try it: What is ∫ x dx from 0 to 4? (Check with the triangle.)
Undo the derivative of x³/3. Raise to 3, divide by 3.
Undo the derivative of x³/3. Raise to 3, divide by 3.
Memory hook. Up one, divide by the new number. 2 → 3, then ÷3.
On the graph. The area under the parabola grows like a cubic — faster than the curve itself.
Watch out. Dividing by the old power (2) instead of the new one (3).
Used for. Volume by slicing: stacking square cross-sections of side x integrates x².
Try it: Differentiate x³/3 + 7 and confirm you get x² back.
The master pattern in reverse: raise the power, divide by the new power.
The master pattern in reverse: raise the power, divide by the new power.
Memory hook. The derivative power rule, rewound. The (n ≠ −1) fine print is where ln x lives.
On the graph. Each accumulation curve grows one power faster than its rate curve.
Watch out. Using it on x⁻¹ — that one's special, and the answer is ln|x|.
Used for. Any power-law rate — physics, economics, biology — accumulates by this one pattern.
Try it: What is ∫ x⁴ dx?
The one power the reverse power rule can't handle gets its own answer: the log.
The one power the reverse power rule can't handle gets its own answer: the log.
Memory hook. n = −1 would force division by zero, so nature invented ln to fill the gap.
On the graph. Accumulating 1/x grows without bound — but logarithmically slowly.
Watch out. Writing x⁰/0. Also: the absolute value matters — ln of a negative isn't real.
Used for. Time to grow between sizes under proportional growth involves ln of the ratio.
Try it: What is ∫ 1/x dx from 1 to e?
eˣ is its own derivative, so it's also its own antiderivative.
eˣ is its own derivative, so it's also its own antiderivative.
Memory hook. The selfie function works in both directions.
On the graph. The accumulated area under eˣ up to x is (almost) the height of eˣ itself.
Watch out. Forgetting +C — even self-similar functions shift.
Used for. Total growth from exponential rates: accumulated interest, accumulated population.
Try it: What is ∫ eˣ dx?
Walk the trig cycle backwards from sin: you land on −cos.
Walk the trig cycle backwards from sin: you land on −cos.
Memory hook. Derivative walks sin → cos forward; integral walks sin → −cos back.
On the graph. Area under one full hump of sine is exactly 2 — try it in the playground.
Watch out. Answering cos x. Differentiate your answer to check: (cos x)′ = −sin x. Wrong sign.
Used for. Net charge from an alternating current is the integral of its sine wave.
Try it: Differentiate −cos x and confirm you get sin x.
Going backwards from cos lands on sin — no minus this time.
Going backwards from cos lands on sin — no minus this time.
Memory hook. One backward step on the cycle: cos came from sin.
On the graph. Where cos is positive, the accumulated sine climbs; where negative, it falls.
Watch out. Adding a stray minus by analogy with ∫sin. Always verify by differentiating.
Used for. Position of an oscillator from its cosine-shaped velocity.
Try it: What is ∫ cos x dx from 0 to π/2?
Out in the world
Pharmacology dosing
A drug clears the bloodstream at a known rate. Integrating that rate tells doctors total exposure over time (the “area under the curve” on a dosing chart — literally an integral), which decides safe dosages.
Common confusion, cleared
“∫x dx = x² — raise the power and done.”
Raise and divide: x²/2 + C. The divide step is what makes differentiating it hand back x.
“The reverse power rule covers every power.”
All but one: x⁻¹ would divide by zero. That power belongs to ln|x| — the exception worth knowing cold.
Check yourself
PracticeQuick check
Match the antiderivative:
Which integral produces ln|x| + C?
Two antiderivatives of the same function always differ by…
Recap
- Reverse power rule: raise the power, divide by the new power, +C.
- ∫1/x dx = ln|x| + C — the exception that proves the rule.
- Always check by differentiating your answer.