Integral of cos(pi*x)/2*(cos(2*n+1)*pi*x)/6 dx

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

   $$\int \frac{\pi x \cos{\left(\pi x \right)} \cos{\left(2 n + 1 \right)}}{2 \cdot 6}\, dx = \frac{\pi \cos{\left(2 n + 1 \right)} \int x \cos{\left(\pi x \right)}\, dx}{12}$$

   1. Use integration by parts:

      $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

      Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\pi x \right)}$.

      Then $\operatorname{du}{\left(x \right)} = 1$.

      To find $v{\left(x \right)}$:

      1. Let $u = \pi x$.

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

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

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

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

            1. The integral of cosine is sine:

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

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

         Now substitute $u$ back in:

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

      Now evaluate the sub-integral.

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

      $$\int \frac{\sin{\left(\pi x \right)}}{\pi}\, dx = \frac{\int \sin{\left(\pi x \right)}\, dx}{\pi}$$

      1. Let $u = \pi x$.

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

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

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

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

            1. The integral of sine is negative cosine:

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

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

         Now substitute $u$ back in:

         $$- \frac{\cos{\left(\pi x \right)}}{\pi}$$

      So, the result is: $- \frac{\cos{\left(\pi x \right)}}{\pi^{2}}$

   So, the result is: $\frac{\pi \left(\frac{x \sin{\left(\pi x \right)}}{\pi} + \frac{\cos{\left(\pi x \right)}}{\pi^{2}}\right) \cos{\left(2 n + 1 \right)}}{12}$

2. Now simplify:

   $$\frac{\left(\pi x \sin{\left(\pi x \right)} + \cos{\left(\pi x \right)}\right) \cos{\left(2 n + 1 \right)}}{12 \pi}$$

3. Add the constant of integration:

   $$\frac{\left(\pi x \sin{\left(\pi x \right)} + \cos{\left(\pi x \right)}\right) \cos{\left(2 n + 1 \right)}}{12 \pi}+ \mathrm{constant}$$

The answer is:

$$\frac{\left(\pi x \sin{\left(\pi x \right)} + \cos{\left(\pi x \right)}\right) \cos{\left(2 n + 1 \right)}}{12 \pi}+ \mathrm{constant}$$