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how is laplace transforms related tointegration?

MIT 18.03SC Differential Equations
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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.)

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