Practice Flashcards
What is integration by substitution undoing?
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All Flashcards in Topic 5.16
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5.16.18 cards
What is integration by substitution undoing?
The chain rule — it's the reverse chain rule. You let u = the inner function so f'(g)·g' dx becomes f'(u) du.
How do you choose u in a substitution?
Let u be the INNER function whose derivative (up to a constant) also appears in the integrand.
After choosing u, how do you replace dx?
Differentiate: du = u' dx, then rewrite u' dx (or dx) in terms of du.
For a DEFINITE integral, what's the clean way to finish?
Change the limits to u-values (put each x-limit into u), then evaluate in u — no switching back.
Find ∫ 2x(x² + 1)⁴ dx.
u = x² + 1, du = 2x dx → ∫ u⁴ du = (x² + 1)⁵/5 + C.
Evaluate ∫₀^(π/2) sin³x cos x dx.
u = sin x, limits 0→1 → ∫₀¹ u³ du = 1/4.
Only a constant factor is missing from u' — what do you do?
Balance it: e.g. if du = 2x dx but you have x dx, then x dx = ½ du. You can pull constants out, never variables.
Find ∫ x√(x² + 3) dx.
u = x² + 3, x dx = ½ du → ½∫ u^(1/2) du = ⅓(x² + 3)^(3/2) + C.
5.16.28 cards
State the integration by parts formula.
∫ u dv = uv − ∫ v du. You differentiate u and integrate dv.
What does LIATE help you decide?
Which factor to call u: Log, Inverse-trig, Algebra (powers of x), Trig, Exponential — first listed becomes u.
When do you use integration by parts (not substitution)?
For a PRODUCT of two different kinds of function (e.g. x·eˣ, x·sin x, x·ln x) where no inner-derivative pair is present.
Find ∫ x cos x dx.
u = x, dv = cos x dx → x sin x − ∫ sin x dx = x sin x + cos x + C.
How is ∫ ln x dx found by parts?
u = ln x, dv = dx (v = x): x ln x − ∫ 1 dx = x ln x − x + C.
How many times must you apply parts for ∫ x² eˣ dx?
Twice — each round drops the power of x by one (x² → x → constant).
Find ∫ x ln x dx.
u = ln x, dv = x dx → (x²/2)ln x − ∫ x/2 dx = (x²/2)ln x − x²/4 + C.
Find ∫ x² eˣ dx.
Parts twice: eˣ(x² − 2x + 2) + C.
Topic 5.16 study notes
Full notes & explanations for Integration by parts (HL only)
Math AA exam skills
Paper structures, command terms & tips
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