Two integrals u-substitution can't touch: products, and rational functions whose
denominators factor. u-sub undoes the chain rule — but products were never the chain rule's
department. This is the heavy machinery.
Two new reversals. Integration by parts runs the product rule backward — it doesn't solve an
integral so much as trade it for a different one, and the skill is trading down. Partial
fractions un-adds a rational function — a fraction whose denominator factors is secretly a sum of
deck-card fractions, and algebra recovers the pieces. Both end, as always, on cards you already own.
Where parts comes from — the product rule, rewound
Same move as last lesson, different rule. Differentiate a product and two terms come
out; integrate that equation and rearrange, and you get a machine for trading one integral for another.
Integrate both sides — the left side collapses by the Fundamental Theorem, and the two right-hand pieces appear:
Rearrange — solve for the piece you were stuck on. This is the formula, and it's a TRADE: you pay ∫u dv and receive uv − ∫v du:
The trade is only worth making if the NEW integral ∫v du is easier — which is the whole question of choosing u and dv
Choosing u — LIATE, and why it works
The trade improves when u gets simpler under differentiation (its job is to
die) and dv is something you can actually integrate. Rank function types by how eagerly they
simplify when differentiated, and you get the classic priority list:
u = the earliest type on the list. Logs and inverse trig differentiate into algebra (ln x → 1/x) — they simplify dramatically, but are painful to integrate. Perfect u material.u dies ↓
dv = the latest type. Exponentials and trig integrate without complaint, forever (eˣ → eˣ). Perfect dv material — and remember dv must swallow the dx.dv recycles ∞
It's a principle, not a spell: u should simplify when differentiated, dv should be integrable. LIATE is just that principle pre-sorted.why > what
Feel the trade — cast it yourself
The formula doesn't care which factor you call u — but the
integral does. Pick a casting and the page runs that round of parts honestly. Try both for each
integral: one trade improves things, the other makes them visibly worse.
That round of parts, run honestly
Pick a casting to see where the trade lands.
The recipe, stepped
press step
The work, shown
Pick an integral
Watch the second one —
LIATE puts the log ahead of the polynomial, which surprises most students. And the third has no
visible product at all… until you invent one.
The tabular method — repeated parts, compressed
When u is a polynomial (it differentiates to zero in a few steps) and dv
integrates forever (eˣ, sin, cos), parts will need several rounds — and every round is the same
bookkeeping. The tabular method runs all the rounds at once: differentiate down the left column until you
hit 0, integrate down the right, attach alternating signs, and multiply along the diagonals. Build
it live:
press step
What each step is doing
When tabular applies
u columnpolynomial → dies at 0
dv columneˣ / sin / cos → forever
signs+ − + − …
read the answeralong the diagonals ↘
Now you build one — ∫ x² sin x dx
Same machine, your hands. Each press asks for the next cell —
and the antiderivative column of sin x is exactly where sign errors breed: sin → −cos → −sin → cos.
The signs in the left column come free; the cells don't.
The cyclic case — when the integral comes back
∫eˣ sin x dx breaks the LIATE worldview: neither factor ever dies. Run parts twice
and something strange happens — the original integral reappears on the right side. That's not
failure; it's the mechanism. Name it and solve for it like any unknown.
press step
The work, shown
Partial fractions — un-adding a fraction
Adding fractions is something you've done since grade school:
1/(x+1) + 2/(x+3) combines into one big fraction. Partial fractions runs that backward: a rational
function whose denominator factors into distinct linear pieces is secretly such a sum, every piece
is the deck's ∫du/u card, and algebra recovers them. This also completes Lesson 3's decision tree — the
"it factors" branch finally has its answer.
The template: one constant over each distinct linear factor. (BC scope: nonrepeating linear factors — that's all the exam asks.)
The cover-up shortcut: to find A, cover its factor in the original and evaluate what's left at that factor's root. Each constant in one line, no systems of equations.
The completed decision tree for any rational integrand:
The cover-up, performed
The trick is named after a physical gesture — so perform it.
Click a factor in the denominator: it gets covered, and the survivors are evaluated at that
factor's root. Each constant falls out in one line.
Click (x + 1) or (x + 3) to cover it.
0 of 2 constants found
What the cover-up computes
Cover a factor to see its constant fall out.
press step
The work, shown
Pick a fraction
The second one is the
full exam gauntlet: top-heavy AND factorable — division first, then partial fractions on the
remainder. Two costumes, removed in order.
Cyclic integrals — when the original returns, name it I and solve algebraically
Partial fractions template for distinct linear factors
The ideas everything else builds on
Parts is a trade, not a solution
∫u dv buys you uv − ∫v du. The trade is good only if the new integral is
easier — if it got worse, your u and dv are backwards.
u dies, dv recycles
Pick u to simplify under differentiation and dv to be integrable. LIATE is
this principle alphabetized — and dv always includes the dx.
Tabular = parts on repeat
Nothing new — just every round of parts at once. Valid exactly when the u-column
reaches zero.
Factor first, always
Rational integrand? Check the denominator: factors → partial fractions;
won't factor → complete the square. And if the numerator is the denominator's derivative, it was a
one-line u-sub all along.
Your turn
Five problems in the exam's voice — including one trap. Try each before revealing.
Quick quiz
Eight fast checks across the whole lesson.
Question 1 of 8
Score: 0 / 0
Common mistakes & exam tips
u and dv backwards
Pick u = eˣ and the new integral is WORSE than the old one. If ∫v du looks harder than what you started with, stop and swap — the formula isn't wrong, the casting is.
dv forgot its dx
dv is a chunk of the integrand INCLUDING dx. Splitting ∫x eˣ dx as u = x, dv = eˣ (no dx) makes v meaningless. Write dv = eˣ dx every time.
Sign slips in the tabular
The signs alternate starting from + and attach to the DIAGONAL products. Writing them next to the derivatives and then multiplying straight across is the classic way to lose every other sign.
Stopping the table early
The tabular column must run until the derivative is exactly 0. Stopping at the constant (instead of differentiating it once more) silently drops the last term.
Partial fractions on the wrong fraction
PF needs a FACTORABLE denominator and a top-light fraction. Top-heavy? Divide first. Won't factor? That's completing the square. Check both before writing A and B.
Panicking in the cycle
When ∫eˣ sin x reappears, students think they went in a circle and start over with different choices — forever. The reappearance IS the solution: name it I, collect, divide.
On the AP exam
Write the casting explicitly on FRQs: “u = x, dv = eˣ dx, du = dx, v = eˣ” — the setup line is a scored step.
Tabular is fully accepted work — but label the columns and show the diagonal products, not just the final answer.
Cover-up is legal and fast for finding A and B, but write one line of evidence: “at x = −1: A = (3(−1)+5)/(−1+3) = 1”.
Before any technique on a rational function, spend five seconds on the denominator: factor? derivative-of-denominator numerator? top-heavy? The diagnosis is worth more than the computation.
The toolkit is complete — every kind of integrand the BC exam can throw now has a technique. One frontier
remains: integrals that run to infinity, or across a point where the function blows up.
Improper Integrals closes the unit — and opens the door to
Unit 10's series. Or head back to the Unit 6 Guide.