The substitution rule, aka integration by substitution, aka u-substitution, is the most common technique of integration. If $F'(x) = f(x)$, then $$ \int f(u) \frac{du}{dx} dx = F(u) + C.$$ By choosing the function $u$ correctly, we can often convert hard integrals into much easier integrals that we know how to evaluate. Unfortunately, there is no magic formula for deciding what $u$ should be -- this is a toolkit, not a recipe.