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sqrt(1 2sqrt(1 3sqrt(1 4sqrt(1 …
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🔥 Day 137 — 2025 MIT Integration Bee (Qualifying) Q12 Evaluate: ∫ sqrt( x * cbrt( x * 4th-root( x * 5th-root( x * … ) ) ) ) dx This “infinite nested radical” looks wild, but it has a self-similar structure. If you call the whole expression y, the nest shows that y = sqrt(x * y). Solve that tiny equation first—then the integral is instant. 👉 Hints • Let y be the entire radical. • Self-similarity ⇒ y = sqrt(x * y) ⇒ y^2 = x y. • Nonzero branch gives y = x (for x ≥ 0). Would you spot the self-sim
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Day 175 — 365 Problems for 365 Days AP Calculus limit with square roots and cancellation: lim x->0 ( sqrt(1 x) - 1 - (1/2)x ) / x^2 This is a classic “designed cancellation” limit. The expression sqrt(1 x) starts with 1 (1/2)x, so subtracting 1 and (1/2)x removes the first two terms and forces you to find the first nonzero term. You can solve it with a short Taylor/binomial expansion, or by rationalizing carefully and simplifying to a clean constant. Project Mentor helps high school students wit
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