Explain how integration via substitution works.

The function in terms of x should be broken down into two easier to integrate functions, like so f(x) = g(h(x)). if we say that u=h(x) is our substitution then we can integrate in terms of u now, however the dx term must also be written in terms of u. for this, differentiate u with respect to x using the function h(x) giving du/dx. now du and dx can be split up and dx can be substituted into the integral in terms of u and du. This can also be done with boundary conditions at the top and bottom of the sigma but it is not necessary as they can be put in when u is converted back in terms of x after the integration.
now the integral with respect to u can be performed. once this is done x can be substituted back in using the relation u=h(x). (this explanation would be aided with a step by step example).