∫ [(8x - 5)/(√(4x² – 5x + 1))] = ...

Jawaban yang benar adalah 2√(4x² – 5x + 1) + C Konsep : Rumus turunan : y = axⁿ → dy/dx = n·a·x⁽ⁿ⁻¹⁾ y = k → dy/dx = 0 Rumus integral : ∫ axⁿ dx = (a/(n+1)) · x⁽ⁿ⁺¹⁾ + C Integral substitusi : Misalkan F(x) adalah anti turunan dari f(x) : ∫ f(g(x)) g'(x) dx = F(g(x)) + C Sifat eksponen : a⁻ⁿ = 1/(aⁿ) ⁿ√(aᵐ) = a^(m/n) Pembahasan : ∫ [(8x - 5)/(√(4x² – 5x + 1))] dx = ∫ [1/(√(4x² – 5x + 1))] · (8x - 5) dx Misalkan : u = 4x² – 5x + 1 du/dx = 2·4x – 5 + 0 du/dx = 8x – 5 du = (8x-5) dx ∫ [1/(√(4x² – 5x + 1))] · (8x - 5) dx = ∫ [1/(√(u))] du = ∫ [1/(u^(1/2))] du = ∫ u^(-1/2) du = (1/(-1/2 + 1)) · u^(-1/2 + 1) + C = (1/(-1/2 + 2/2)) · u^(-1/2 + 2/2) + C = (1/(1/2)) · u^(1/2) + C = 1 · (2/1) · √(u) + C = 2√(4x² – 5x + 1) + C Jadi nilai dari ∫ [(8x - 5)/(√(4x² – 5x + 1))] dx = 2√(4x² – 5x + 1) + C