Find the Error: Integration by Parts (part 1)

It is a beautiful Spring day. You leave your calculus class feeling sad and depressed. You aren’t sad because of the class itself. On the contrary, you have just learned an amazing integration technique: Integration by Parts. You aren’t sad because it is your birthday. On the contrary, you are still young enough to actually be happy about it. You are sad because you know that every time you learn something really wonderful in calculus, a wild-eyed stranger runs up to you and shows you a “proof” that it is false. Sure enough, as you cross the street, he is waiting on the other side.

“Good morning, Sonny,” he says.

“I just learned integration by parts. Let me have it.”

“What do you mean?” he asks.

“Aren’t you going to run around telling me that all of math is lies?”

“Well, if you insist, he chuckles…” and he hands you this piece of paper:

“Hey,” you say, “I don’t get it! You did everything right this time!”

“Yup!” says the hungry looking stranger.

“But… zero isn’t equal to negative one!”

“Nope!” he says.

You didn’t think he could pique your interest again, but he has. Spite him. Find the error in his reasoning.

  1. Find the error: Differentiation
  2. Find the error: L’Hopital’s Rule
  3. Find the error: Related Rates
  4. Find the error: Fundamental Theorem of Calculus
  5. Find the error: U-substitution
  6. Find the error: Integration by Parts (part 1)
  7. Find the error: Integration by Parts (part 2)
  8. Find the error: Trigonometric Integration
  9. Find the error: Taylor Series
  10. Find the error: Improper Integrals and Taylor Series
  11. Find the error: Separation of Variables
  12. Find the error: Separation of Variables – Exponential Growth
  13. Find the error: Solutions
  14. Find the error: Credits