CramX
RegisterLog in
Back to AI Flashcard MakerMathematics /AP Calculus AB: 9.2.1 Undoing the Chain Rule

AP Calculus AB: 9.2.1 Undoing the Chain Rule

Mathematics9 CardsCreated 3 months ago

This content introduces integration by substitution, a technique used to reverse the chain rule during integration. It explains how to recognize integrals involving composite functions and their derivatives, provides step-by-step examples, and outlines the conditions under which substitution is effective. The focus is on simplifying complex integrals by identifying patterns formed by the chain rule.

PrintEmbedImportReport

Undoing the Chain Rule

  • Since differentiation and integration are inverse operations, some of the patterns used when differentiating can be seen when working with integrals.

  • One method for evaluating integrals involves untangling the chain rule. This technique is called integration by substitution.

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/9

Key Terms

Term
Definition

Undoing the Chain Rule

  • Since differentiation and integration are inverse operations, some of the patterns used when differentiating can be seen when working with ...

note

  • Here are some warm-up problems.

  • Remember, to find the derivative of a composite
    function you must use the chain rule.

    ...

Evaluate.

∫7(x^3−1)^6(3x^2)dx

(x^3−1)^7+C

To determine if an integral is a good candidate for integration by substitution:

The integral must be made up of a composition of functions and the derivative of the inside function

Evaluate.

∫5(x^2+1)4(2x)dx

(x^2+1)^5+C

Which of the following integrals is not a good candidate for integration by substitution?

∫xsinxdx

Log in to view all terms

Related Flashcard Decks

Mathematics

Essential SAT Math Formulas

This deck covers key formulas and concepts essential for the SAT Math section, including slope, distance, and vertex forms.

10 cards
View Deck
Mathematics

Algebra 1 Unit 1 Mastery Test

These properties of equality help maintain balance in an equation when performing the same operation on both sides. Inverse operations, like using addition to undo subtraction, are key tools in solving equations.

29 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 2

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

53 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 1

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

55 cards
View Deck
Mathematics

Unit 2 Progress Check: MCQ Part A

To determine the point where the tangent line to the graph of the function 𝑓f has a slope of 2 for 𝑥 >0 x>0, we use the fact that the slope of the tangent line at any point is given by the derivative 𝑓′(𝑥)f′(x).

15 cards
View Deck
Mathematics

Customary Liquid Measurement Units

This set of flashcards outlines key relationships between customary units of liquid measurement, including pints, cups, quarts, and gallons. It provides conversion facts to help understand how these units relate to one another in the U.S. measurement system.

3 cards
View Deck

Study Tips

  • Press F to enter focus mode for distraction-free studying
  • Review cards regularly to improve retention
  • Try to recall the answer before flipping the card
  • Share this deck with friends to study together
AP Calculus AB: 9.2.1 Undoing the Chain Rule
TermDefinition

Undoing the Chain Rule

  • Since differentiation and integration are inverse operations, some of the patterns used when differentiating can be seen when working with integrals.

  • One method for evaluating integrals involves untangling the chain rule. This technique is called integration by substitution.

note

  • Here are some warm-up problems.

  • Remember, to find the derivative of a composite
    function you must use the chain rule.

  • Notice that the derivative is the product of a
    composite function and the derivative of the inside.

  • This derivative is the product of a composite
    function, another composite function, and the
    derivative of the inside of the second composite
    function.

  • Here is a trick question. You could solve this
    indefinite integral by multiplying everything out and
    working it term by term. However, there is an easier
    way.

  • Notice that the integrand is equal to one of the
    derivatives you found above. So you already know
    a function that produces this integrand as its
    derivative. Since that is what integration finds, that
    means you already know the integral.

  • When you see a composite function multiplied by its
    derivative in the integrand, it is a good hint that you
    can use a technique to evaluate the integral called
    integration by substitution.

Evaluate.

∫7(x^3−1)^6(3x^2)dx

(x^3−1)^7+C

To determine if an integral is a good candidate for integration by substitution:

The integral must be made up of a composition of functions and the derivative of the inside function

Evaluate.

∫5(x^2+1)4(2x)dx

(x^2+1)^5+C

Which of the following integrals is not a good candidate for integration by substitution?

∫xsinxdx

Evaluate.

∫2sinxcosxdx

sin^2x+C

Integration by substitution (also called change of variable) is a way to undo which of the following?

The chain rule.

An integral is solvable by integration by substitution if and only if the integrand can be expressed as which of the following?

g′(h (x)) · h′(x)