2nd/2ways: count perm Show P(n,2)+P(n,1)=n^2

4th/5ways: triangular num: Show t(n-1) + tn = n^2 where tn = nth triangular number

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trig Write each expression as an equivalent algebraic expression involving only x.

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2nd way: integral ∫ log √x(x^n) dx

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1st way: exp ∫(e^x)/x (1+x log(x)) dx

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MP 2000 Hindustan: Evaluate ∫(tan(x)+cot(x))^2 dx

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CBSE 2000 Hindustan: integral (1+tan(x))/(x+log(sec(x)) dx

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Solve x(x-1)dy/dx - (x-2)y=x^3(2x-1) via Integrating Facto[IF]

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3rd/3part: Maclaurin Series [ahp kia]

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1st/3part: Maclaurin Series [ahp kia]

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2024 #3 MIT Integration Bee Final Lightning Round

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AP Calculus 4th way: exp ∫1/(1+e^x) dx via 3rd u-sub

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logic Idempotent law show x=xVx

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Suppose E & F are mutually exclusive events. Must E & F be complementary? Explain your answer.

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trig integral a) Show that sin(t) cos(t)=1/2 sin^2(t)+c

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Lamar U: trig integral ∫cos^5(2t) sin^2(2t) dt

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5th/10parts: Material implication

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trig integral sin(tan(x)) cos(tan(x)) sec^2(x) dx [chain rule in reverse]

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2 ways: Find ∫(ax^x+bx+c) dx

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logic: tautology p→q≡(¬q→¬p)

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2nd way: integral ∫1/(a + √√b (x^2)) dx

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