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

To tackle the integral \( \int 7(\cosh(x))^3 \, dx \), we can start by pulling out the constant: \[ 7 \int (\cosh(x))^3 \, dx. \] Next, we can use the identity for hyperbolic functions to rewrite \((\cosh(x))^3\). We can express \((\cosh(x))^3\) in terms of other hyperbolic functions or use integration by parts or its reduction formula. The reduction can get a bit technical, but for a straightforward approach, the integral can be simplified using the known integral for hyperbolic functions: \[ \int (\cosh(x))^n \, dx \text{ can typically involve polynomials and exponential functions.} \] Thus, with the correct application and integral tables, we can compute: \[ \int (\cosh(x))^3 \, dx = \frac{1}{3}(\cosh(x) \sinh(x) + 2x) + C. \] Inserting this back into our equation gives: \[ 7 \left( \frac{1}{3}(\cosh(x) \sinh(x) + 2x) \right) + C. \] Therefore, we have: \[ \frac{7}{3}(\cosh(x) \sinh(x) + 2x) + C. \]