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ayudenme con esta integral por sustitucion trigonometrica.?

ayuda necesito su ayuda para resolver esta intergral por el metodo de sustitucion trigonométrica.

∫ dx/ (a^2 + X^2)^5/2 si la respuesta a llegar es x / a^2 √(a^2 + x^2) + C

me urge y necesito de su ayuda se los agradeceria mucho.

1 Answer

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  • Anonymous
    1 decade ago
    Favorite Answer

    ∫ dx/ (a^2 + X^2)^5/2

    x = atgθ; dx = asec²θdθ

    ∫ asec²θdθ/√[a²+ (atgθ)²]⁵

    ∫ asec²θdθ/√[a²(1+tg²θ)]⁵

    ∫ asec²θdθ/[a⁵√(sec²θ)⁵]

    ∫ sec²θdθ/[a⁴sec⁵θ]

    1/a⁴ ∫ dθ/sec³θ = 1/a⁴ ∫ cos³θdθ

    1/a⁴ ∫ cos²θcosθdθ

    1/a⁴ ∫ (1-sen²θ)cosθdθ

    1/a⁴ ∫ [cosθdθ-3sen²θcosθdθ/3]

    1/a⁴ [senθdθ-sen³θ/3]

    Hipotenusa =(a^2 + X^2)^(1/2)

    senθ = x/[a(a^2 + X^2)^(1/2)]

    1/a⁴ {x /[a√(a²+x²)] - x³ / [3a³√(a²+x²)³]}

    x /[a⁵√(a²+x²)] - x³ / [3a⁷√(a²+x²)³]

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