How would you antidifferentiate the given integral?
Or change of variable u-substitution where u=x-3, du=dx, and x=u+3?
What if you take a step back and practice look for and make use of structure? (Just in case you didn’t previously note the structure.)
Could decomposing the expression into an equivalent form help us antidifferentiate?
I’ve shown this problem to students for at least 20 years. This year, one of my students shared the structure he saw, which makes antidifferentiating easy compared to change of variable u-substitution (and quicker than long division).
Learning mathematics using the Standards for Mathematical Practice changes our perspective. The ordinary problems that I’ve been posing for over 20 years become extraordinary because we are learning to view problems with the lens of reason abstractly and quantitatively, look for and make use of structure, use appropriate tools strategically, look for and express regularity in repeated reasoning, … The Math Practices are becoming our starting point for interacting with a problem. The Math Practices are becoming habits of mind for my students and for me.
And even better, the Math Practices aren’t just about math. One of my calculus students looked up at our Math Practices posters in the front of the room last week and noted, “Those are just good practices for life in general. I think they threw in ‘model with mathematics’ to make it sound more like math.” And so the journey continues, and it gets better with every class period …