1. The problem is to find the integral of $\cosh\left(\frac{x}{a}\right)$ with respect to $x$. 2. Recall that the integral of $\cosh(u)$ with respect to $u$ is $\sinh(u) + C$. 3. Here, let $u = \frac{x}{a}$, so $du = \frac{1}{a} dx$ or $dx = a \, du$. 4. Substitute into the integral: $$\int \cosh\left(\frac{x}{a}\right) dx = \int \cosh(u) \cdot a \, du = a \int \cosh(u) du$$ 5. Integrate: $$a \int \cosh(u) du = a \sinh(u) + C$$ 6. Substitute back $u = \frac{x}{a}$: $$a \sinh\left(\frac{x}{a}\right) + C$$ 7. Therefore, the integral is: $$\int \cosh\left(\frac{x}{a}\right) dx = a \sinh\left(\frac{x}{a}\right) + C$$