Toggle the table of contents

Quiz:Integration by parts: Difference between revisions

Quiz
Discussion

English

Read
View source
View history

Tools

Actions

Read
View source
View history
Refresh

General

What links here
Related changes
Special pages
Printable version
Permanent link
Page information

Help

From Calculus

@@ Line 7: / Line 7: @@
 - Suppose <math>F</math> and <math>G</math> are everywhere differentiable. Given an expression for the antiderivative for the [[pointwise product of functions]] <math>F'G</math>, we can obtain an expression for the antiderivative for the pointwise product <math>FG'</math>.
 - If <math>F</math> is a one-to-one function, we can find an antiderivative for <math>F^{-1}</math> in terms of <math>F^{-1}</math> and an antiderivative for <math>F</math>.
+{Which of the following correctly describes the relationship between integration by parts and integration by u-substitution when deciding to integrate a [[pointwise product of functions]] or a [[composite of two functions]]?
+|type="()"}
+- Integration by parts, which is obtained from the [[product rule for differentiation]], is the exclusive strategy for integrating products. Integration by u-substitution, which is obtained from the [[chain rule for differentiation]], is the exclusive strategy for integrating composites.
+- Integration by parts, which is obtained from the [[product rule for differentiation]], is the exclusive strategy for integrating composites. Integration by u-substitution, which is obtained from the [[chain rule for differentiation]], is the exclusive strategy for integrating products.
++ Both methods are useful for both types of integrations. Specifically, integration by parts helps with certain kinds of products and composites, and integration by u-substitution helps with certain kinds of products.
+|| For instance, for products of the form math>h(g(x))g'(x)</math> it is useful to integrate by the u-substitution <math>u = g(x)</math>. On the other hand, for function such as <math>x \sin x</math>, we use integration by parts. Even for pure composites, we may use integration by parts, either directly or combined with integration by u-substitution. For instance, <math>\cos (\ln x)</math>
+{Which of the following is an ''incorrect'' way of applying integration by parts twice?
+|type="()"}
+- After applying integration by parts once, we get a new product. Choose as the ''part to integrate'' the factor in the product arising from integration, and as the ''part to differentiate'' the factor in the product arising from differentiation.
++ After applying integration by parts once, we get a new product. Choose as the ''part to differentiate'' the factor in the product arising from integration, and as the ''part to integrate'' the factor in the product arising from differentiation.
+|| See [[Integration by parts#Circular trap]]. Basically, if we do this, we get back to where we started from and have no new information.
+- Neither method is incorrect in general. The first method is used for straightforward integrations and the second method is used for the [[recursive version of integration by parts]].
+|| No! The recursive version of integration by parts also differentiates again the part obtained by differentiation, but ''additionally'' it may perform some manipulation based on algebraic or trigonometric identities.
+{Which of the following integrations can be done ''without'' integration by parts, and purely using [[integration by u-substitution]] and the knowledge of the antiderivative of the cosine function?
+|type="()"}
+- <math>x^2 \cos x</math>
+- <math>x \cos^2 x</math>
++ <math>x \cos (x^2)</math>
+- <math>x^2 \cos (x^2)</math>
+- <math>x^2 \cos^2 x</math>
+</quiz>
+==Choosing the parts to integrate and differentiate==
+<quiz display=simple>
+{Suppose <math>f</math> is a function with a known antiderivative <math>F</math>. Which of the following is correct (and can be deduced using integration by parts)?
+|type="()"}
+- Knowledge of an antiderivative for <math>x \mapsto f(x^2)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.
+- Knowledge of an antiderivative for <math>x \mapsto xf(x^2)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.
+- Knowledge of an antiderivative for <math>x \mapsto x^2f(x^2)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.
+- Knowledge of an antiderivative for <math>x \mapsto x^2f(x)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.
++ Knowledge of an antiderivative for <math>x \mapsto x^2f(x)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.
 </quiz>

Revision as of 23:51, 28 December 2011

Key observations

1 Which of the following is not true (the ones that are true can be deduced from integration by parts)?

	We can compute an expression for the antiderivative of the pointwise product of functions $f g$ based on knowledge of expressions for $f$ , $g$ , and their antiderivatives.
	Suppose $F$ and $G$ are everywhere differentiable. Given an expression for the antiderivative for the pointwise product of functions $F^{'} G$ , we can obtain an expression for the antiderivative for the pointwise product $F G^{'}$ .
	If $F$ is a one-to-one function, we can find an antiderivative for $F^{- 1}$ in terms of $F^{- 1}$ and an antiderivative for $F$ .

2 Which of the following correctly describes the relationship between integration by parts and integration by u-substitution when deciding to integrate a pointwise product of functions or a composite of two functions?

	Integration by parts, which is obtained from the product rule for differentiation, is the exclusive strategy for integrating products. Integration by u-substitution, which is obtained from the chain rule for differentiation, is the exclusive strategy for integrating composites.
	Integration by parts, which is obtained from the product rule for differentiation, is the exclusive strategy for integrating composites. Integration by u-substitution, which is obtained from the chain rule for differentiation, is the exclusive strategy for integrating products.
	Both methods are useful for both types of integrations. Specifically, integration by parts helps with certain kinds of products and composites, and integration by u-substitution helps with certain kinds of products.

3 Which of the following is an incorrect way of applying integration by parts twice?

	After applying integration by parts once, we get a new product. Choose as the part to integrate the factor in the product arising from integration, and as the part to differentiate the factor in the product arising from differentiation.
	After applying integration by parts once, we get a new product. Choose as the part to differentiate the factor in the product arising from integration, and as the part to integrate the factor in the product arising from differentiation.
	Neither method is incorrect in general. The first method is used for straightforward integrations and the second method is used for the recursive version of integration by parts.

4 Which of the following integrations can be done without integration by parts, and purely using integration by u-substitution and the knowledge of the antiderivative of the cosine function?

	$x^{2} \cos x$
	$x \cos^{2} x$
	$x \cos (x^{2})$
	$x^{2} \cos (x^{2})$
	$x^{2} \cos^{2} x$

Choosing the parts to integrate and differentiate

Suppose $f$ is a function with a known antiderivative $F$ . Which of the following is correct (and can be deduced using integration by parts)?

	Knowledge of an antiderivative for $x \mapsto f (x^{2})$ is equivalent to knowledge of an antiderivative for $F$ .
	Knowledge of an antiderivative for $x \mapsto x f (x^{2})$ is equivalent to knowledge of an antiderivative for $F$ .
	Knowledge of an antiderivative for $x \mapsto x^{2} f (x^{2})$ is equivalent to knowledge of an antiderivative for $F$ .
	Knowledge of an antiderivative for $x \mapsto x^{2} f (x)$ is equivalent to knowledge of an antiderivative for $F$ .
	Knowledge of an antiderivative for $x \mapsto x^{2} f (x)$ is equivalent to knowledge of an antiderivative for $F$ .

Retrieved from "https://calculus.subwiki.org/w/index.php?title=Quiz:Integration_by_parts&oldid=857"