BrainTok
ENESIT

Calculus II and Multivariable Calculus: Integration techniques · Substitution and Integration by Parts

How substitution and integration by parts undo the chain and product rules, and how to choose the right one for an integral.

Calculus II and Multivariable Calculus · 27 questions

Try 3 questions

1. u-substitution is essentially which differentiation rule run in reverse?

  1. The product rule
  2. The chain rule
  3. The quotient rule
Show the answer

Answer: The chain rule

The chain rule says the derivative of F(g(x)) is F'(g(x))·g'(x). Substitution spots that pattern inside an integrand and undoes it.

2. Integration by parts comes from reversing which differentiation rule?

  1. The product rule
  2. The power rule
  3. The chain rule
Show the answer

Answer: The product rule

Integrating both sides of (uv)' = u'v + uv' and rearranging gives the parts formula. It trades one integral for another that is hopefully easier.

3. What is the integration by parts formula for ∫u dv?

  1. uv − ∫v du
  2. uv + ∫v du
  3. ∫u du · ∫v dv
Show the answer

Answer: uv − ∫v du

From the product rule, ∫u dv = uv − ∫v du. The minus sign matters: forgetting it is one of the most common errors.

The rest of the quiz, a few questions at a time

BrainTok asks a few questions from quizzes like these before the apps you choose open, so your screen time turns into learning. For iPhone.

Join the beta

More Mechanical Engineering quizzes