How to Memorize Integration Formulas (Calc and A-Level)
The best way to handle how to memorize integration formulas is to realize most of them are derivatives you already know, read backwards. Learn the handful of core integrals (power, 1/x, eˣ, the trig basics), memorize the few genuinely new ones (like ∫tan x and ∫sec x), use LIATE for integration by parts, and drill them all with spaced flashcards. Then practice choosing the method, because that's what the test really asks.
If you keep second-guessing the signs on trig integrals, FlashDeck turns your formula list into cards and brings back exactly the ones you keep missing: Build my study plan.
Same content whether you call it Calc 2, AP Calc AB/BC, A-level Maths (UK) or first-year calc in Canada.
Integrals are derivatives in reverse
Integration (antidifferentiation) undoes differentiation. So every derivative you know is secretly an integral:
- d/dx [sin x] = cos x → ∫cos x dx = sin x + C
- d/dx [tan x] = sec²x → ∫sec²x dx = tan x + C
- d/dx [arctan x] = 1/(1 + x²) → ∫1/(1 + x²) dx = arctan x + C
If your derivatives aren't solid yet, fix that first; it's half the work: how to memorize derivatives.
The "differentiate to check" habit
Unsure about an integral? Differentiate your answer. If you get back the original function, you're right. This one habit catches most sign errors and turns doubt into a ten-second check.
The core integration formulas
These are the basic rules of integration. Every "+ C" matters on indefinite integrals, so don't drop it.
| Integral | Result | Note |
|---|---|---|
| ∫xⁿ dx | xⁿ⁺¹/(n + 1) + C | n ≠ −1. "Add one, divide by the new power" |
| ∫1/x dx | ln|x| + C | the n = −1 case |
| ∫eˣ dx | eˣ + C | its own integral too |
| ∫aˣ dx | aˣ/ln a + C | flip of d/dx [aˣ] = aˣ ln a |
| ∫k·f(x) dx | k∫f(x) dx | constants come out |
| ∫(f ± g) dx | ∫f dx ± ∫g dx | split sums |
A-level extras worth carding
A-level Maths specs lean on a few patterns:
- ∫f′(x)/f(x) dx = ln|f(x)| + C (top is the derivative of the bottom)
- ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / (a(n + 1)) + C (divide by the inner derivative)
- ∫e^(ax) dx = (1/a)e^(ax) + C
That "divide by a" pattern is reverse chain rule, and it covers a surprising number of exam questions.
Trig integrals
This is where how to memorize integrals of trig functions gets real.
The direct flips (from derivatives)
| Integral | Result |
|---|---|
| ∫sin x dx | −cos x + C |
| ∫cos x dx | sin x + C |
| ∫sec²x dx | tan x + C |
| ∫csc²x dx | −cot x + C |
| ∫sec x tan x dx | sec x + C |
| ∫csc x cot x dx | −csc x + C |
Sign trick: integrating sin gives a negative, and so do the co- ones. Or just differentiate to check: d/dx [−cos x] = sin x. (checks out)
The four you actually have to memorize
These don't come from a basic derivative flip, so they need cards:
- ∫tan x dx = ln|sec x| + C (or −ln|cos x| + C)
- ∫cot x dx = ln|sin x| + C
- ∫sec x dx = ln|sec x + tan x| + C
- ∫csc x dx = −ln|csc x + cot x| + C
Hook for tan and cot: tan = sin/cos, which is "−(derivative of bottom) over bottom," so it's a log. Cot = cos/sin, derivative on top, so it's ln|sin x|.
sin² and cos²
You can't integrate these directly. Use power-reducing identities first:
- ∫sin²x dx = x/2 − sin(2x)/4 + C
- ∫cos²x dx = x/2 + sin(2x)/4 + C
Those come from sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2. If those identities are fuzzy: how to memorize trig identities.
Integration by parts: LIATE
The integration by parts rule:
∫u dv = uv − ∫v du
A classic way to remember it: "ultraviolet voodoo" (uv − ∫v du). Silly, and it sticks.
Choosing u with LIATE
Pick u as whichever type comes first in this list:
- Logarithmic (ln x)
- Inverse trig (arctan x)
- Algebraic (x², x)
- Trig (sin x)
- Exponential (eˣ)
Whatever's left is dv. Example: ∫x eˣ dx. Algebraic beats exponential, so u = x, dv = eˣ dx. Then du = dx, v = eˣ, and the answer is x eˣ − eˣ + C.
LIATE is a guideline, not a law. Some UK teachers use a shorter version like LATE, same idea. The goal is to pick a u that gets simpler when you differentiate it.
Classic trap: ∫ln x dx. There's no second function, so set u = ln x and dv = dx. Answer: x ln x − x + C.
Keeping 20-plus integrals straight, and knowing which ones you'd actually get right under pressure, is tough to judge on your own. Paste your integration formulas into FlashDeck and it makes one card per formula from what you pasted. Tap Knew it or Missed it; the misses come back tomorrow and the solid ones space out, about ten minutes a day: Build my study plan.
Drill formulas with cards
Card fronts that work:
- Q: ∫sec x dx? A: ln|sec x + tan x| + C
- Q: Integration by parts formula? A: ∫u dv = uv − ∫v du
- Q: LIATE order? A: Log, Inverse trig, Algebraic, Trig, Exponential
- Q: ∫1/(1 + x²) dx? A: arctan x + C
- Q: ∫f′(x)/f(x) dx? A: ln|f(x)| + C
Add a few cards that name a situation, not a formula: "Q: ∫x cos x dx: which method? A: By parts, u = x."
Practice choosing the method
Formulas are the easy part. Integration tips and tricks mostly boil down to this: recognize what kind of integral you're looking at.
A quick decision list:
- Is it a basic formula? Just use it.
- Can you simplify first? Expand, split fractions, or use a trig identity.
- Is there a function and its derivative? Try u-substitution.
- Is it a product of two different types? Try by parts with LIATE.
- Is it a rational function? Partial fractions (Calc 2 / A-level).
Mix these up in practice; doing ten u-sub problems in a row teaches you u-sub, not how to spot it. Three to five mixed problems a day in the run-up to your test works better than one massive session.
Walk into the integrals unit with the formulas already locked
Paste your integration formulas into FlashDeck and in seconds you get a deck built only from what you pasted. It shows only the cards you're about to forget, so the formulas stay automatic while your practice time goes to choosing the method.
FAQ
What is the easiest way to learn integration?
Start by treating integrals as derivatives in reverse, and always check your answer by differentiating it. Memorize the short list of integrals that don't come from simple flips, and do a few mixed problems each day.
What integration rules do I need for A-level?
The core ones: powers, 1/x, eˣ, sin and cos, f′(x)/f(x), (ax + b)ⁿ, plus integration by substitution and by parts. Check your exam board's formula booklet, since some results are provided and don't need memorizing.
How do I memorize integrals of trig functions?
Most are derivatives read backwards, so learn the trig derivatives first. Then card the four that aren't simple flips: ∫tan x, ∫cot x, ∫sec x and ∫csc x. Differentiate to check your signs.
What is the LIATE rule in integration by parts?
LIATE is an order for picking u: Logarithmic, Inverse trig, Algebraic, Trig, Exponential. Choose the type that appears first as u and the rest as dv. It's a guideline for making the new integral simpler.
Is there a list of integration rules I can study from?
Yes, your textbook or formula sheet has one, and the tables above cover the basics. Don't just reread the list; turn it into flashcards and quiz yourself, since recall is what makes formulas stick.