Determine o valor da integral ∫ 8 1 4 u 8 + U 2 8 √ u − 2 u 2

Determine o valor da integral ∫

8

1

4

u

8

+

U

2

8

√

u

−

2

u

2

Para resolver essa integral, podemos utilizar a substituição trigonométrica. Fazemos a substituição u = 2secθ, e então temos: du/dθ = 2secθ.tanθ.dθ √(u²-4) = √(4sec²θ-4) = 2tanθ Substituindo na integral, temos: ∫(8,1/4) u^8/(u²-4)^(1/2) du = ∫(θ1,θ2) 2secθ.tanθ.(2secθ)^8/(2secθ)^2.2tanθ dθ = 2^7 ∫(θ1,θ2) sec^6θ dθ Podemos integrar sec^6θ utilizando a fórmula de redução de potência, que nos dá: ∫ sec^6θ dθ = (1/5)sec^4θ.tanθ + (4/5)∫ sec^2θ dθ - (2/5)∫ sec^4θ dθ Substituindo de volta para u, temos: ∫(8,1/4) u^8/(u²-4)^(1/2) du = 2^7 [(1/5)sec^4θ.tanθ + (4/5)tanθ - (2/5)∫ sec^4θ dθ] + C Para encontrar os limites de integração θ1 e θ2, podemos utilizar a relação u = 2secθ e substituir os valores de u = 8 e u = 1/4, obtendo: θ1 = cos^-1(1/4) θ2 = cos^-1(8/2) Substituindo na fórmula acima, temos: ∫(8,1/4) u^8/(u²-4)^(1/2) du = 2^7 [(1/5)(1/4)^4.tan(cos^-1(1/4)) + (4/5)tan(cos^-1(8/2)) - (2/5)∫ sec^4(cos^-1(u/2)) du] + C Calculando os valores numéricos, temos: ∫(8,1/4) u^8/(u²-4)^(1/2) du ≈ 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