Many students are turned off by problems in calculus before they’ve even begun. It just looks intimidating. Images from the media, like the one below, may evoke panic. It may also remind you of a time when you’ve gotten lost midway through a math class: all of the lines start to blur together.
Though it is not easy, tackling a calculus problem methodically can help us untangle this puzzle. So, take a deep breath, and let’s get into it.
Consider the following integral:
The Greek letter phi alone may seem formidable, but we could name it anything!
Is that better?
The Reframe
Trying to solve this problem directly doesn’t work. If you manage to do it, you’ll almost certainly make a careless mistake or two by omitting steps.
Instead, how can we rewrite this problem as a simpler one that we can solve?
Let’s change our frame of reference by making a substitution. We could use the variable u by convention, but again, any variable will do.
Here are a couple of options:
1. 
2. 
We’ve reached a fork in the road.
Multiple Paths
When presented with a fork in the road, take both! When we make the above substitutions, we get the following integrals:
1. 
2. 
Both are mathematically valid. Which one would you rather solve?
I’d choose the second. Now we just need to solve the second integral. Goodbye, trig and hello, power rule.
The Toolbox
Calculus is generally split into semesters: one on differentiation and one on integration. By the time you’re getting to integrals like this, you may already be familiar with the basics of derivatives.
But here’s the thing: integration is really just differentiation in reverse. At the end of the day, there are very few new tools needed to solve integration problems. It is not a memorization game. The key is to practice applying the existing tools in your toolbox in new ways. Active studying through solving problems is always more effective than memorizing specific procedures.
Calculus and “real life”
As a calculus teacher, I was never under the pretense that my students would be faced with a trig integral problem in their future careers. Even in fields like engineering, which rely heavily on calculus, integral calculators like Wolfram Alpha exist for routine tasks. Moreover in applied math, approximate answers are often sufficient.
AP Calculus can feel contrived at times. When am I ever going to use this in real life, you might ask?
To this, I say: real life will be full of problems that you can’t easily solve. You will need to reframe them, and you will need to explore different paths forward. Mathematics is a training ground for building the problem-solving skills needed in any domain.
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