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La integral de \( 3 \cosh^2(x) \) es \( \frac{3x}{2} + \frac{3 \sinh(2x)}{4} + C \), donde \( C \) es una constante.
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Para resolver la integral
∫ 3 cosh²(x) dx
podemos utilizar la identidad hiperbólica:
cosh²(x) = (1 + cosh(2x)) / 2.
Entonces, sustituyendo en la integral:
∫ 3 cosh²(x) dx = 3 ∫ cosh²(x) dx
= 3 ∫ [1 + cosh(2x)]/2 dx
= (3/2) ∫ [1 + cosh(2x)] dx.
Ahora separamos la integral:
(3/2) [∫ 1 dx + ∫ cosh(2x) dx].
La integral de 1 con respecto a x es:
∫ 1 dx = x.
Para la integral de cosh(2x), usamos el cambio de variable considerando que la derivada de sinh(2x) es 2 cosh(2x). Por ello:
∫ cosh(2x) dx = (1/2) sinh(2x).
Sustituyendo ambas integrales:
(3/2) [x + (1/2) sinh(2x)] = (3x/2) + (3 sinh(2x)/4).
Por lo tanto, la integral es:
∫ 3 cosh²(x) dx = (3x/2) + (3 sinh(2x)/4) + C,
donde C es la constante de integración.
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Simplify this solution Beyond the Answer
To integrate \(3 \cosh^2(x)\), recall the identity \(\cosh^2(x) = \frac{1 + \cosh(2x)}{2}\). Thus, you can rewrite the integral as follows: \[ 3 \int \cosh^2(x) \, dx = 3 \int \frac{1 + \cosh(2x)}{2} \, dx = \frac{3}{2} \left( \int 1 \, dx + \int \cosh(2x) \, dx \right) \] Now, integrating each term gives: \[ \int 1 \, dx = x, \quad \text{and} \quad \int \cosh(2x) \, dx = \frac{1}{2} \sinh(2x) \] Putting it all together, the integral becomes: \[ \frac{3}{2} \left( x + \frac{1}{2} \sinh(2x) \right) + C \] This simplifies to: \[ \frac{3x}{2} + \frac{3}{4} \sinh(2x) + C \] So, the final result is: \[ \int 3 \cosh^2(x) \, dx = \frac{3x}{2} + \frac{3}{4} \sinh(2x) + C \]
