Preparing for Success in Calculus II

All through Calculus I students learn about derivatives and the process of taking derivatives.

Finding an integral, the main topic of Calculus II, is essentially the opposite operation as taking a derivative.

For example, if the derivative of f(x) is f'(x), then the integral of f'(x) will be f(x) + C where C is an arbitrary constant.

We have to use this arbitrary constant because the derivative of a constant is zero. To clarify this point, if we take the derivative of 2x + 4 and the derivative of 2x + 10, we'll get the answer of 2 either way.

Then to take the integral of 2, we'll have 2x + C, since there's no way for us to figure out what the constant was at the end of the function. Besides being the opposite of taking a derivative, integrals are an extremely powerful mathematical tool on their own, and is one of the most important mathematical discoveries ever. Essentially, an integral as a function allows you to easily find the size of certain objects that are exceptionally difficult to measure with algebra. For example, imagine the function f(x) = x² - 4, and look at the area under the x-axis but above the curve.

The shape this makes is not a standard shape that we know how to find the area of from geometry or using algebraic methods, but with calculus and the use of an integral, you can find this area exactly inside of three or four steps.

This is the power of the integral and why the majority of Calculus II is dedicated to it.

Just like with derivatives there are a handful of rules and guidelines you learn to make your life easier, there are a number of methods for taking integrals that you'll have to learn to become skilled at integration. Two of these rules that are first introduced in Calculus II that seem to give students the most trouble are using partial fractions and integration by parts.
Integration by parts in particular is difficult for students coming straight out of Calculus I because while the process of taking derivatives is mostly a straight-forward mechanical one, integration requires much more strategy and less specific linear-style thinking.

The point I want to make here is that most of your frustration when dealing with integration by parts will come from a lack of experience with integration, not because you don't understand the material.

On the other hand, integration by partial fractions is a much more mechanical method like the methods used in taking derivatives. Even though it's mechanical in nature, the main trouble students have with partial fractions is that it's a very simple process taught in the middle of a lot of other fairly complicated processes, so students tend to try to make it a lot harder and complicated than it needs to be.
Using the process of partial fractions turns your integration problem into a very straight-forward logic problem where you're trying to find a few constants (A, B, C,.
.

.) in an equation of one variable.

For example, you could have something like 6 = A(x-2) + Bx for all x.

You could let x = 2, and that will quickly give you that the constant B = 3.
Since you're left with 6 = A(x-2) + 3x, if you let x = 3 now you get 6 = A + 9, and A = -3.
This simple example illustrates your main tool when finding these constants: manipulating the value of x.