This is a calculus problem that arose in an AP Physics class. The student said that his teacher didn't know how to integrate a particular expression without referring to a table of integrals. The solution is a straightforward example of trig substitution.
I never said or implied that x is constant. The original integral is a function of two variables: f(x,y). The two variables are independent (i.e., x cannot be written as a function of y), so x behaves like a constant in the example because we are integrating
with respect to y. Calculus students should be comfortable integrating over a variable other than x.
Would you mind uploading the .tex file you used to create this pdf. I'm doing a a project for calculus and am not sure how to create the right triangle in latex.
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