2 ways: Σ 1? where k=m to n

1st/4 ways: integral ∫1/(x + √x) dx

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Review: Linear form, Quadratic form & Bilinear form

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1st way: exp ∫(e^x)/x (1+x log(x)) dx

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2nd way: integral ∫ log √x(x^n) dx

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MP 98 Hindustan: Evaluate (i) ∫e^loge(x) dx (ii) ∫e^(-loge(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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Review Add Math: 1.2/parts: Show a^m a^n = a^(m+n) where m & n are any 2 positive integers

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5 ways: integral ∫a^(px+q) dx

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#6 2019 MIT Integration Bee Final

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exp Differentiate (i) b^e^x (ii) b^x&e (iii) (b^e)^x (iv) (b^x)^e

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Screen Recording 2023-12-02 at 3.42.17 AM

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4 ways: exp integral ∫a^(bx+c) dx

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2nd/2 parts: 2 (elementary) rules

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2nd way: derivative ∫log (f(x))^g(x) (p(x))^r(x)

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Alternate way: ∫(5+2x)/(x+x^4)dx

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2nd way: integral ∫x^k ln(x) dx = x^(k+1)/(k+1) ln(x) - x^(k+1)/(k+1)^2 + C where C ≠ -1

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2nd/2parts example Show Σk^2 = ΣΣmin(i,j) where i,j,k=1 to n

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integral ∫√(x-a)/x dx

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Bijection of elements in tn & a pair of elements of (n+1)

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