All through Calculus I, students will learn about derivatives and the process of adopting derivatives. The search for the integral, the main theme of calculus II, in fact, is the opposite operation, like taking a derivative. For example, if the derivative f (x) is equal to f (#), then the integral of f ^ (x) will be f (x) + C, where C is an arbitrary constant. We must use this arbitrary constant, since the derivative of the constant is zero. To clarify this point, if we take the derivative of 2x + 4 and the derivative of 2x + 10, we will get answer 2 in any case. Then, to take the integral of 2, we have 2x + C, since we cannot find out that the constant was at the end of the function.
Contrary to taking a derivative, integrals are an extremely powerful mathematical tool in their own right and are one of the most important mathematical discoveries. In fact, the integral as a function allows you to easily find the size of certain objects that are extremely difficult to measure using algebra. For example, imagine the function f (x) = x²-4 and look at the area under the x-axis, but above the curve. The form it makes is not a standard form, that we know how to find a field of geometry or use algebraic methods, but with calculus and using the integral you can find this area exactly within three or four steps. This is the power of the integral and why most of calculus II is devoted to it.
As in the case of derivatives, there are several rules and recommendations that you learn to make your life easier, there are a number of methods for making integrals that you will need to learn in order to become qualified in integration. Two of these rules, which were first introduced in Calculus II, which seem to give students a big problem, are partial fractions and integration in parts. Integration in parts is particularly difficult for students who go straight from Calculus I, because, although the process of adopting derivatives is basically mechanical, integration requires a much larger strategy and less specific thinking in a linear style. The point I want to make here is that most of your frustration when working with integration in parts will be due to a lack of integration experience, and not because you don’t understand the material.
On the other hand, integration with partial fractions is a much more mechanical method, similar to the methods used in making derivatives. Despite the fact that they are mechanical in nature, the main problems of students with partial fractions is that a very simple process is taught in the middle of many other rather complex processes, so students tend to do this much more complicated and more complicated than necessary. Using the fractional fraction process turns your integration problem into a very straightforward logical problem, in which you try to find several constants (A, B, C, ...) in the equation of one variable. For example, you might have something like 6 = A (x-2) + Bx for all x. You can enable x = 2, and this will quickly give you this constant B = 3. Since you are left with 6 = A (x-2) + 3x, if you are now x = 3, you will get 6 = A + 9 and A = -3. This simple example illustrates your basic tool when searching for these constants: manipulating the x value.
