3 Important Methods of Integration To Know
U-Substitution, Integration by Parts, Trigonometric Substitution
One of the most essential parts of Calculus, integration details the techniques of finding an antiderivative, F(x), of the function f(x). It has many applications both in mathematics and physics that involve volume, area, and mechanics. In this article, we’ll examine three of the key methods for integration to have in your arsenal.
Integration is essentially the opposite of differentiation. The variety of difficulty within the subject of integration is extremely extensive, with some problems taking 5 seconds to solve and some problems taking 5 minutes. For example, we know that the derivative of x³ is 3x². Therefore, when we integrate 3x², we get x³+C. It is always important to remember the +C, since the derivative of a constant is always zero, so any constant value can be tacked on to the end of a derivative expression and differentiating will yield the same result.
However, not all problems are as simple as finding the integral of 3x². What is the integral of sin(2x)? One may jump to write down -cos(2x) + C, but sadly, this would be incorrect. The reasoning behind why this solution is incorrect leads us into our first method of integration: U-Substitution.
As I was taught, U-Substitution is a way of dealing with the chain rule from differentiation: It reverses it! The chain rule deals with derivatives of composite functions. In examples like this, we say that the derivative of the function f(g(x)) is f’(g(x))*g’(x). This is why the derivative of -cos(2x) isn’t just sin(2x): we are missing an extra factor of 2 from the derivative of the inside function 2x. Now it should be apparent to us why integrating sin(2x) doesn’t simply yield -cos(2x). It is absolutely necessary to “account” for the chain rule in both differentiation and integration problems. Let’s look at an example problem together.
Our first step should be to locate any familiar functions and derivatives, going term by term down the integral. Because we are reversing the chain rule, we want to set our u = sin(x), leaving our du = cos(x)dx.