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Expanding the product gives 8 sec²x − 32 sec x tan x, and both terms are standard antiderivative table entries. The most general antiderivative is F(x) = 8 tan x − 32 sec x + C, which returns the original integrand when differentiated.
The arc length integral collapses here: y′ = x√(x² − 2) makes 1 + (y′)² = (x² − 1)², so on [2, 4] the integrand is just x² − 1 and the length is ∫₂⁴ (x² − 1) dx = 50/3 ≈ 16.667.
Polar arc length is ∫√(r² + (dr/dθ)²) dθ. For r = 1 − sin θ the radicand reduces to 2 − 2 sin θ, a half-angle identity turns it into a sine, and the exact value of the integral is 8.
Counting atoms on both sides of each candidate equation leaves only 2H₂ + O₂ → 2H₂O with equal hydrogen and oxygen counts, so that is the correctly balanced equation.
The valid Lewis dot symbol for each element's monatomic ion follows directly from its valence electron count: the metals Na, Mg, Ca, K and Sr lose electrons and show no dots, while Cl, Br and F each gain one electron and are drawn with a full octet.
Classifying 1-hexyne, 3-octyne, cyclooctyne and propyne shows that 3-octyne and cyclooctyne are internal alkynes: their triple-bond carbons are bonded to carbon on both sides, with no hydrogen on either sp carbon.
A C₆H₁₂ hydrocarbon that takes up one equivalent of H₂ over palladium and cleaves to propanoic acid plus acetone must be 2-methylpent-2-ene; hydrogenating its double bond gives 2-methylpentane.
Choose the complete aerobic-yeast respiration answer: acetyl-CoA enters the citric acid cycle, NADH feeds the electron transport chain, and the item's approximate total is 32 ATP per glucose.
The organ level combines two or more different tissue types for a specific function; it sits above tissue and below organ system in human structural organization.