This week I started back to my journey in mathematics, this time tackling Calculus II. Starting out the week was the section I ended with in Calc I, integration by parts. No sweat! However, the second section for the week was Trigonometric Integrals. I’m not certain what I was expecting exactly going into the section, however, after working a few problems I realized something very interesting: I really hadn’t learned any new skills, what I had learned was a way to leverage skills I already had to solve problems that were previously unapproachable. Let me demonstrate with finding a solution for the following integration problem:
What am I doing here?
I’ve learned that the asking and answering of this question for myself is critical to solving problems. In this case, I’m trying to solve an integration problem but, truth be told, I know of know way to integrate this problem as it is. It’s not in a basic form. The only option in this circumstance is to manipulate the equation until it resembles a basic form that can be solved. So if I had to take a wild guess as to what I’m really doing here, it would be that I’m going to be manipulating this equation to get it into a basic form.
Manipulating to integrate
To this point I’ve learned a few ways to manipulate problems:
- Algebraic - Perhaps the simplest manipulation, manipulating the problem algebraically to simplify it into a basic form that can be solved.
- U-Substitution - Useful for solving integrals that are composed of a function and its derivative.
- Integration by Parts - Leveraging a proven relationship to break apart an integral into a function and a, hopefully, simpler integral to solve.
Note that at times, these manipulation strategies can be combined in order to achieve a solution. However, in the case of the problem I am trying to solve, none of these situations directly apply.
It should come as no surprise then, that all there is to solving trigonometric integrals is leveraging trigonometric manipulations to get an equation into a basic, solvable, form.Continue reading...