lim n→∞ n∫tanⁿ(x) dx [0, π/4] MIT Integration Bee 2022 Quarterfinal #1 Problem 2

Описание: #mitintegrationbee #trigonometry #integrationbysubstitution

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Questions of MIT Integration Bee
https://math.mit.edu/~yyao1/pdf/

Following are some of playlist and ideos on calculus, algebra, logarithm, trigonometry and some older video on ioT/Smart Home:

MIT Integration Bee
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Area Under Curves
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JEE-Main, India
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UK AQA/CIE A-Level
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Stanford Math Tournament
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Calculus
[1] Solving Gaussian Integral (integration of gaussian function) by Feynman’s Method.
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[2] Created another 1 minute video for Happy π day! Solve an integral and the answer is π!
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[3] Solving integral x^(9)*(x^(10) + 1)/(x^(20) + 1) dx. Can we solve it using substitution?
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[4] Solving integral 1/(-sqrt(x^2 + 4*x + 3)) dx. Can it be solved by substitution?
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[5] Solving sum of k=0 to ∞ for integral sin^(2k) (x) dx. Solution is much easier than expected!
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[6] Solving area of region bounded between 2 curves, i.e., {(x,y): x^2 ≤ y ≤ x + 2} using integration.
   • Solving area of region bounded between 2 c...  
[7] Solving integral (x^2 - 1)/(x^4 - 1) dx.
   • ∫(x²- 1)/(x⁴ - 1) dx. Solve an interesting...  
[8] Given y=kx^2, x=ky^2, k is +ve constant & area enclosed between parabola curves is 1 unit^2. Find k.
   • Given y=kx^2, x=ky^2, k is +ve constant & ...  
[9] Given f(x) = integral (x^5-16x)/(x^8-256) dx and f(-2) = π/32. Find f(2). Explanation in 60 seconds

   • f(x) = ∫ (x^5-16x)/(x^8-256) dx and f(-2) ...  
Algebra
[1] Is this a quartic or quadratic problem? Can we solve it within 60 seconds and without calculator?
   • Solving sqrt(100*102*102*104 + 4). Is this...  
[2] Solving (10016*104*12*8 + 256)^0.5 but without using calculator & computer. Answer is a nice number!
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[4] 15 seconds video on solving sqrt(10300 - 9*10*10*11) without using calculator and computer.
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[5] Given x^2 - x - 1 = 0, solve cos^1 ((3/2)*(7*x + 1)/(x^9 - 18*x - 13)).
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[6] An infinite radical equation, i.e., sqrt(3*sqrt(7*sqrt(…))) solve by Geometric Series.
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[7] If (√ 2 + √ 3)^11 = a √ 2 + b√ 3, then (a² - (3/2)*b²)^11 = ?
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[8] Given (|x-3| - 4)^2 = 4. How many answers for x?
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Logarithm/exponential
[1] Solving a finite power equation of (ln(x))^2. The solution is surprisingly easier than expected.
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[2] Given that ln(3) is approximately equal to 1.099, find ln(3.1) using calculus without calculator.
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[3] Solving x^(ln x) = x for real numbers. How many answers for x?
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[4] Solving a radical-absolute equation. Is this a quadratic equation too?
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[5] Solving a logarithm-quadratic question of Stanford Math Tournament 2021. Calculator is not allowed.
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[6] 15 seconds video to explain the answer for ln(-1) based on principal value of trigonometry.
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[7] 15 seconds video on solving x^(ln|x|) = 1/|x|. There are 4 answers for x?
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Trigonometry
[1] Solving a trigonometry-power equation? Solution is fairly simple and can be explained in 60 seconds.
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[2] 45 seconds video deriving cos(-θ)=cos(θ) & sin(-θ)=-sin(θ). Basic identities of trigonometry Part 5.
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[3] 15 seconds video for deriving sec^2(x) = tan^2(x) + 1. Basic identities of trigonometry Part 1.
   • 15 seconds video for deriving sec^2(x) = t...  
[4] 15 seconds video for deriving sin^2(x) + cos^2(x) = 1. Basic identities of trigonometry Part 2.
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IoT/Smart Home
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[2] Can We Remote Access Node-red Server from another computer? Part 1: Wireless/Wire LAN
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[3] How to install Amazon Alexa App on iPhone in Malaysia or region without it in Apple Store (Feb 2021)
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[4] Smart Home: Security Automation Alarm System using Aqara G2H Camera Hub, Motion & Contact Sensors
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[5] Node-red Iot Server & Telegram bot - Part 2: Search & display photo using telegrambot (Windows 10)
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