# Find the Error: Trigonometric Integration

It is a beautiful Spring morning. Everywhere you look, people are happily going to their classes, or coming from their classes. “College is fun!” calls out one student, and about twenty more yell “Sure is!” in unison. Someone else calls out, “I love History!” A bunch of other students call “Great subject!” in response. Swept up in the spirit of things, you call out, “Calculus is wonderful!” “Lies! Lies!” calls out one familiar voice. You wheel around and directly behind you is a wild-eyed hungry-looking stranger.

“Oh, don’t be silly,” you say. “I just learned about trigonometric integration. It wasn’t that hard a section, and there isn’t a single lie in it.”

He looks up at you and says, “Oh, really? Perhaps you can take a quick true-or-false test, and see how easy the section is.” The stranger then whips out a sheet of paper with this on it:

“Both are clearly true!” he shouts. And you realize that he is correct. His first integral is a simple u substitution (Let u = 2x) as is the second (Let u = cos(x)). He continues: “AND we know that -2sin(2x) = -4sin x cos x!” You realize that this is also correct because sin(2x) = 2sin(x)cos(x). “Thus cos 2x=2cos2x! Ho ho!”