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🔥 Day 121 — 2025 MIT Integration Bee (Qualifying) Q4 Evaluate: Integral from 1 to e^e of [ log( x^(log(x^x)) ) / x^2 ] dx Looks wild, but the trick is to simplify before integrating. If you collapse the nested powers/logs the integrand turns into something that begs for a one-line u-sub. 👉 Hints • Use: log(a^b) = blog(a) and log(x^x) = xlog(x). • You should get (log x)^2 / x, then let u = log x. Would you spot it under time pressure? ⏱️ 💭 Drop your answer method below 👇 ✅ Full step-by-step s
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🔥 Day 117 — Can you solve this? Evaluate the integral (a > 0, b ≥ 0): ∫₀^∞ e^(−a x) · sin(bx) / x dx It looks tough — but a parameter derivative makes it collapse 📉 Differentiate with respect to b, use the Laplace transform of cos(bx), then integrate back. 👉 Hint: F′(b) = ∫₀^∞ e^(−a x) cos(bx) dx = a/(a² b²). Integrate in b and use F(0) = 0 to nail the constant 👀 Would you land the exact expression before time runs out? ⏱️ 💭 Drop your answer below 👇 ✅ Full step-by-step solution is already
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To evaluate this integral, let u=x 5. That means, the denominator becomes u³. While x=u-5,then substitute the value of u in the numerator and simplify before integrating each term. #viral_video #usubstitution #integration #calculus #fypviralシ
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