Question

how is laplace transforms related tointegration?

Answer 1

Can you be more specific about what your question is? The Laplace transform of a function is an integral. By definition,
\[(Lf)(s)=\int\limits_{0^{-}}^{t}f(t)e ^{-st}dt\]
for all values of s for which the integral converges.

So, if you want to find the Laplace transform of a function, one method would be to use the definition, and calculate an integral. (Another method is to use a table, along with theorems like the convolution theorem, and translation theorems.)