I'm confused about differentiation and integration, could you explain these to me?

Let's start with differentiation. Differentiation is the instantaneous rate of change. We differentiate functions to find their rate of change as a function of their input values, this is known as a function's derivative. We are familiar with finding the gradient, or rate of change, of straight lines, well differentiation can be thought of as the gradient of a function at any particular point. In order to compute the derivative of a function, we have a number of rules which are applied according to the nature of the function we wish to differentiate. One such rule for dealing with terms of a polynomial is "multiply by the power and lower the power by one".
Integration is process of computing the integral of a function. An integral is an infinite sum of infinitely small things which lie in an interval. This is equivalent to finding the area underneath a curve between to points. It is one of the deepest facts in Mathematics, and hence is known as the fundamental theorem of calculus, that integration is the reverse process of differentiation - to find the integral of a function f we need to find a function g whose derivative is equal to f. If the interval is known (and hence the limits are given) we compute this integral at the two limits and subtract the two. If the interval is not known, we need to think of the most general possible function g which differentiates to f. Practically, this simply means adding an unknown constant to our specific function.