Integral of cos^2(5x) dx

1. Rewrite the integrand:

   $$\cos^{2}{\left(5 x \right)} = \frac{\cos{\left(10 x \right)}}{2} + \frac{1}{2}$$

2. Integrate term-by-term:

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \frac{\cos{\left(10 x \right)}}{2}\, dx = \frac{\int \cos{\left(10 x \right)}\, dx}{2}$$

      1. Let $u = 10 x$.

         Then let $du = 10 dx$ and substitute $\frac{du}{10}$:

         $$\int \frac{\cos{\left(u \right)}}{100}\, du$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \frac{\cos{\left(u \right)}}{10}\, du = \frac{\int \cos{\left(u \right)}\, du}{10}$$

            1. The integral of cosine is sine:

               $$\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$$

            So, the result is: $\frac{\sin{\left(u \right)}}{10}$

         Now substitute $u$ back in:

         $$\frac{\sin{\left(10 x \right)}}{10}$$

      So, the result is: $\frac{\sin{\left(10 x \right)}}{20}$

   1. The integral of a constant is the constant times the variable of integration:

      $$\int \frac{1}{2}\, dx = \frac{x}{2}$$

   The result is: $\frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}$

3. Add the constant of integration:

   $$\frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}$$

The answer is:

$$\frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}$$