9.2.1 Undoing the Chain Rule Flashcards Preview

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Flashcards in 9.2.1 Undoing the Chain Rule Deck (9)
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1
Q

Undoing the Chain Rule

A
  • 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.
2
Q

note

A
  • 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.
3
Q

Evaluate.

∫7(x^3−1)^6(3x^2)dx

A

(x^3−1)^7+C

4
Q

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

A

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

5
Q

Evaluate.

∫5(x^2+1)4(2x)dx

A

(x^2+1)^5+C

6
Q

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

A

∫xsinxdx

7
Q

Evaluate.

∫2sinxcosxdx

A

sin^2x+C

8
Q

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

A

The chain rule.

9
Q

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

A

g′(h (x)) · h′(x)

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