Integral of p*i*0*cos(5*x+1)*dx dx

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

   $$\int 0 i p \cos{\left(5 x + 1 \right)}\, dx = 0$$

   1. Let $u = 5 x + 1$.

      Then let $du = 5 dx$ and substitute $\frac{du}{5}$:

      $$\int \frac{\cos{\left(u \right)}}{5}\, du$$

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

         $$\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$$

         1. The integral of cosine is sine:

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

         So, the result is: $\frac{\sin{\left(u \right)}}{5}$

      Now substitute $u$ back in:

      $$\frac{\sin{\left(5 x + 1 \right)}}{5}$$

   So, the result is: $0$

The answer is:

$$0+ \mathrm{constant}$$