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🔥 Day 152 — 2026 MIT Integration Bee (Qualifying) Q18 365 Problems for 365 Days Evaluate: Integral of (sin^2(x)/x^2 − sin(2x)/x) dx This one is a “recognize the derivative” speed problem. The integrand looks messy, but it’s literally the derivative of a super simple expression if you think “quotient rule trig identity.” If you try to integrate directly, it’s pain. If you spot the structure, it’s instant. Hints: \t•\tLook for something like (sin^2 x)/x \t•\tDifferentiate it with the quotient rul
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🔥 Day 133 — 2025 MIT Integration Bee (Qualifying) Q17 Evaluate: ∫ sin(x) * sinh(x) dx Wild mix of circular and hyperbolic functions. The fastest path is a classic double “integration by parts” loop: set up I = ∫ sin(x) sinh(x) dx, integrate by parts twice, and watch I appear on both sides so you can solve for it. 👉 Hints • First parts: u = sin x, dv = sinh x dx. • Second parts (on the new integral): u = cos x, dv = cosh x dx. • You’ll end with 2I = (sin x) cosh x − (cos x) sinh x. Would you ca
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🔥 Day 128 — 2025 MIT Integration Bee (Qualifying) Q11 Evaluate: Integral from 0 to 20 of floor(x)^2 dx This is a classic step-function trap. Don’t try to “anti-differentiate” floor(x). Instead, split the interval into unit chunks and add rectangles. Once you see the pattern, it turns into a clean sum of squares. 👉 Hints • On [k, k 1), floor(x) = k. • Integral becomes sum_{k=0}^{19} k^2. • Use the formula for 1^2 2^2 … n^2. Would you spot the step-function strategy fast enough? ⏱️ 💭 Comment yo
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Day 153 — 2026 MIT Integration Bee Qualifying (Q19) 365 Problems for 365 Days Today’s problem is a nested logarithm integration question that shows up a lot in AP Calculus and contest math: you’re looking for the “perfect substitution” hidden in the denominator xlog(x). If you choose the right u-sub, the integral turns into a clean integration-by-parts problem with ulog(u). Problem: Integral of [ log(log x) * log(log(log x)) ] / [ x * log x ] dx High-yield hint: \t•\tIf you see dx/(x log x), thi
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