Question

Método de sustitución​

Answer 1

Respuesta:

(arctan(√4x²-9)/3))/6 + C

Explicación paso a paso:

∫(1/2x√(4x²-9))dx = ½∫(1/(x√4x²-9)) dx = ...

sustituye: u = √4x² - 9 → du/dx = 4x/√4x²-9 → dx = √4x²-9/4x du

... =(½) ∫ (1/(x√4x²-9)) • (√4x²-9/4x du) = ...

... = ∫ (1/(u² + 9)) du , vuelve a sustituir: v = u/3 → dv/du = ⅓ → du = 3 dv

→ ∫(3/(9v² + 9) dv = ⅓ ∫ (1/v² + 1) dv =...

... = (⅓)• (arctan (v)) → ⅓ ∫(1/v²+1) dv = (arctan (v))/3 → quitamos lo sustituido: v = u/3

(arctan(u/3))/3 → quitamos lo sustituido: u = √4x² - 9

→ ½∫(1/(x√4x²-9)) dx = ½ (arctan((√4x² - 9)/3))/3 = ...

... = arctan((√4x²-9)/3))/6 + C