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2nd/2 parts: Mean 1+2+3+...+n is just (1+n)/2. Why?
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1st way: exp ∫(e^x)/x (1+x log(x)) dx
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AI 2008 Hindustan: Evaluate integral ∫ 2/3 cos(x)/sin^2(x) dx
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MPPET 1996 Hindustan: trig integral ∫ sin^2(x) -cos^2(x)/(sin^2(x) cos^2(x)) dx
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MP 8,93,96,97,99 Hindustan: Evaluate ∫ sec(x)/(sec(x)-tan(x)) dx
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MP 2000 Hindustan: Evaluate ∫(tan(x)+cot(x))^2 dx
2nd way: integral ∫ log √x(x^n) dx
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Screen Recording 2024-03-03 at 12.03.07 AM
2 ways: integral (3+5x)/(x(x-1)) dx
2nd/3 ways: How many chords can be drawn through 21 points on a circle?
1st/2 parts stat: SLR ∑ŷk = ∑yk where k=1,n
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2nd/2ways: Taylor series of sin(x)
derivative If y=e^(a arc sin(x)), prove that (1-x^2)y2 - xy1 = a^2y.
2nd way: integral ∫1/(√(px-q) - √(px+r)) dx
Screen Recording 2024-05-19 at 9.47.07 PM
derivative: If x=cos(log(y)), show that (1-x^2)y2 - xy1 = y
We lost original x's but still know mean x̄ & n. Now we collected m new observations.
trig integral ∫sin(2x) cos(2x) / (9-cos^4(2x)) dx
2nd way: trig integral ∫sec^(3/2)(x) tan^3(x) dx
Solve cos^2(x) dy/dx +y = tan(x) via Integrating factor[IF]