9.2.1 Undoing the Chain Rule Flashcards
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)
