A direct application of integration is the solution of an ordinary differential equation of type \[\frac{\dd y}{\dd t}=f(t)\] for some function \(f(t)\). The general solution of this first order ODE can be found by integration: \[y(t)=\int f(t)\,\dd t\tiny.\]
A stronger suggestion of integration comes from rewriting the differential equation in terms of differentials: \[\dd y=f(t)\,\dd t\tiny.\] The variables \(y\) and \(t\) are now separated from each other: \(\dd y\) is to the left and everything with \(t\) in it to the right. Integrating the left- and right-hand sides yields: \[\int \dd y=\int f(t)\,\dd t\quad \text{i.e.}\quad y = \int f(t)\,\dd t\tiny.\] The integration constants separately do not provide new solutions and can therefore be combined into a single integration constant. This is in the final formula already done secretly.
The first-order ODE \[\frac{\dd y}{\dd t}=3t^2\] is in differential form \[\dd y=3t^2\,\dd t.\] The right-hand side of this equation can also be written as \(\dd(t^3)\), because \(t^3\) is an antiderivative of \(3t^2\). So we have an equality between two differentials: \[\dd y=\dd(t^3).\] The functions ‘behind the d's’ are thus equal to each other up to a constant: \[y=t^3+c\] for some constant \(c\).
You may also apply integration to the left- and right-hand side in \[\dd y=3t^2\,\dd t\] Then you get \[\int \dd y=\int 3t^2\,\dd t\] and the left- and right-hand side are equal to \(y\) and \(t^3\), respectively, up to an integration constant.
When you have an initial value problem \[\frac{\dd y}{\dd t}=f(t),\quad y(t_0)=y_0\] for a certain function \(f(t)\) and parameters \(t_0\) and \(y_0\), then you can also work with definite integrals in order to determine the solution of the initial value problem: \[\int_{y_0}^{y(t)}d\eta=\int_{t_0}^t f(\tau)\,d\tau\] So: \[y(t) = y_0+ \int_{t_0}^t f(\tau)\,d\tau\]