Integral of cos(2*pi*x)f(t) dy

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

   $$\int f t \cos{\left(2 \pi x \right)}\, dx = f t \int \cos{\left(2 \pi x \right)}\, dx$$

   1. Let $u = 2 \pi x$.

      Then let $du = 2 \pi dx$ and substitute $\frac{du}{2 \pi}$:

      $$\int \frac{\cos{\left(u \right)}}{4 \pi^{2}}\, du$$

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

         $$\int \frac{\cos{\left(u \right)}}{2 \pi}\, du = \frac{\int \cos{\left(u \right)}\, du}{2 \pi}$$

         1. The integral of cosine is sine:

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

         So, the result is: $\frac{\sin{\left(u \right)}}{2 \pi}$

      Now substitute $u$ back in:

      $$\frac{\sin{\left(2 \pi x \right)}}{2 \pi}$$

   So, the result is: $\frac{f t \sin{\left(2 \pi x \right)}}{2 \pi}$

2. Add the constant of integration:

   $$\frac{f t \sin{\left(2 \pi x \right)}}{2 \pi}+ \mathrm{constant}$$

The answer is:

$$\frac{f t \sin{\left(2 \pi x \right)}}{2 \pi}+ \mathrm{constant}$$