\( \int _{}^{}7(\cosh (x))^{3} d x \)

To evaluate the integral \( \int 7(\cosh(x))^3 \, dx \), we can use a hyperbolic identity to simplify our work. Recall that \( \cosh^3(x) = \frac{3}{4} \cosh(x) + \frac{1}{4} \cosh(3x) \). Using this identity, the integral becomes: \[ \int 7(\cosh(x))^3 \, dx = 7 \left( \int \frac{3}{4} \cosh(x) \, dx + \int \frac{1}{4} \cosh(3x) \, dx \right). \] After calculating the integrals separately, we have: \[ \int \cosh(x) \, dx = \sinh(x) \quad \text{and} \quad \int \cosh(3x) \, dx = \frac{1}{3} \sinh(3x). \] Thus, putting it all together, we find: \[ = 7 \left( \frac{3}{4} \sinh(x) + \frac{1}{12} \sinh(3x) \right) + C. \] So the final answer is: \[ \frac{21}{4} \sinh(x) + \frac{7}{12} \sinh(3x) + C. \]