9.2.1 Undoing the Chain Rule Flashcards Preview

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

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.

2

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.

3

Evaluate.
∫7(x^3−1)^6(3x^2)dx

(x^3−1)^7+C

4

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

5

Evaluate.
∫5(x^2+1)4(2x)dx

(x^2+1)^5+C

6

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

∫xsinxdx

7

Evaluate.
∫2sinxcosxdx

sin^2x+C

8

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

The chain rule.

9

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)

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