Integration by substitution
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Flip to reveal answersWhat is integration by substitution undoing?
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All 8 Flashcards — Integration by substitution
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Question
What is integration by substitution undoing?
Answer
The chain rule — it's the reverse chain rule. You let u = the inner function so f'(g)·g' dx becomes f'(u) du.
Question
How do you choose u in a substitution?
Answer
Let u be the INNER function whose derivative (up to a constant) also appears in the integrand.
Question
After choosing u, how do you replace dx?
Answer
Differentiate: du = u' dx, then rewrite u' dx (or dx) in terms of du.
Question
For a DEFINITE integral, what's the clean way to finish?
Answer
Change the limits to u-values (put each x-limit into u), then evaluate in u — no switching back.
Question
Find ∫ 2x(x² + 1)⁴ dx.
Answer
u = x² + 1, du = 2x dx → ∫ u⁴ du = (x² + 1)⁵/5 + C.
Question
Evaluate ∫₀^(π/2) sin³x cos x dx.
Answer
u = sin x, limits 0→1 → ∫₀¹ u³ du = 1/4.
Question
Only a constant factor is missing from u' — what do you do?
Answer
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.
Question
Find ∫ x√(x² + 3) dx.
Answer
u = x² + 3, x dx = ½ du → ½∫ u^(1/2) du = ⅓(x² + 3)^(3/2) + C.
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Full study notes for Integration by substitution
Topic 5.16 hub
Integration by parts (HL only)
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