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Answer
La integral de sinh(4x - 5) dx es (1/4) cosh(4x - 5) + C.
Solution
Para resolver la integral ∫ sinh(4x - 5) dx, realizamos el cambio de variable:
1. Sea u = 4x - 5, de donde se sigue que du/dx = 4, lo que implica que dx = du/4.
2. Sustituyendo en la integral:
∫ sinh(4x - 5) dx = ∫ sinh(u) (du/4) = (1/4) ∫ sinh(u) du.
3. La integral de sinh(u) es cosh(u), es decir, ∫ sinh(u) du = cosh(u) + C.
4. Por lo tanto, la solución es:
(1/4) cosh(u) + C.
5. Reemplazamos u por 4x - 5:
(1/4) cosh(4x - 5) + C.
Así, el resultado final es:
(1/4) cosh(4x - 5) + C.
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Simplify this solution Bonus Knowledge
To solve the integral \( \int \sinh ( 4 x - 5 ) d x \), you can use a simple substitution. Let \( u = 4x - 5 \). Then, \( du = 4dx \) or \( dx = \frac{du}{4} \). Changing the variable in the integral, we can write: \[ \int \sinh(u) \frac{du}{4} = \frac{1}{4} \int \sinh(u) \, du \] The integral of \( \sinh(u) \) is \( \cosh(u) + C\). Thus, we have: \[ \frac{1}{4} (\cosh(u) + C) = \frac{1}{4} \cosh(4x - 5) + C \] So the final answer is: \[ \int \sinh(4x - 5) \, dx = \frac{1}{4} \cosh(4x - 5) + C \]
