∫x(1/2 + ln(x))ln(ln(x)) dx = ? MIT Integration Bee 2025, Regular Season, Problem 9. #integrationbee
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∫(2√x + 1)/√(x² + x√x ) dx = ? MIT Integration Bee 2026, Regular Season, Problem 7.
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∫(2√x + 1)/√(x² + x√x ) dx = ? MIT Integration Bee 2026, Regular Season, Problem 7.
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🔥 Day 143 — 2026 MIT Integration Bee (Qualifying) Q9 Evaluate: ∫ x^2 sin(x) dx Classic double integration-by-parts. Set u = x^2 so the power drops on the next pass, then do parts again on ∫ x cos x. Clean, mechanical, and fast once you’ve practiced it. 👉 Hints • 1st parts: u = x^2, dv = sin x dx • 2nd parts on ∫ x cos x • Expect a combo of x sin x and cos x terms 📣 Project Mentor: free guides for AP self-study, competitions, and research/internships—see my bio for more. 💭 Drop your method be
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🔥 Day 124 — 2025 MIT Integration Bee (Qualifying) Q7 Evaluate: ∫ [ x*log(x)cos(x) − sin(x) ] / [ xlog(x)^2 ] dx This looks nasty… until you reorganize it. Rewrite the integrand so the structure screams “reverse quotient rule.” If you can spot the right function in the numerator/denominator pairing, the antiderivative pops out immediately. 👉 Hints • Rearrange as: [ log(x)*cos(x) − (sin(x)/x) ] / [ log(x)^2 ]. • Ask: whose derivative has that exact pattern? Would you recognize it under speed-run
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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 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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