What is an example of horizontal integration?

What happened in the integration in this example problem?

• this is an example problem in my textbook a_n = 1/pi integral cos(nx)dx integrating from 0 to pi = 0 for n /= 0 = 1 for n = 0 <-- ?? I feel like I'm missing ...show more

• Answer:

  if n = 0 then cos( n theta) = 1 at all times Remember that an integral is the inverse of differentiation. 1 dx = x [ this is because d x^n / dx = x^(n-1) and if n = 1 then dx^1 / dx = 1 hence the integral of 1 dx = x^1 ] so cos(0) dx from 0 to pi = pi hence 1/ pi * cos(0) dx from 0 to pi = 1/ pi * pi = 1. ( inadvertently used sin in my original response but sin(0) = 0 at all times NOT 1 as I quoted)