Quiz:Equivalence of integration problems - Revision history

Vipul: /* General functions= */

2024-03-18T22:06:36Z

General functions=

← Older revision		Revision as of 22:06, 18 March 2024
Line 1:		Line 1:
	This quiz considers questions about how one integration problem can be converted to another using [[integration by parts]] and [[integration by u-substitution]].		This quiz considers questions about how one integration problem can be converted to another using [[integration by parts]] and [[integration by u-substitution]].

	==General functions===		==General functions==

	<quiz display=simple>		<quiz display=simple>

Vipul: /* Specific functions */

2012-02-20T03:49:58Z

Specific functions

@@ Line 80: / Line 80: @@
 - <matH>1/(ab)</math> is an integer
 - <math>a/b</math> is an integer
 </quiz>

Vipul at 03:43, 20 February 2012

2012-02-20T03:43:51Z

@@ Line 1: / Line 1: @@
 <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)?
@@ Line 48: / Line 52: @@
 - An antiderivative for <math>x \mapsto f(1/\sqrt{x})</math>, domain positive reals.
 </quiz>

Vipul: Created page with " {Suppose $f$ is a function with a known antiderivative $F$ . Which of the following is correct (and can be deduced using integration..."

2012-02-20T03:42:07Z

Created page with "<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..."

New page

<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 xf(x)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.

{Suppose <math>f</math> is a function with a known antiderivative <math>F</math>. Which of the following integration problems is ''not'' equivalent to the others?
|type="()"}
- <math>\int f(\sqrt{x}) \, dx</math>
- <math>\int xf(x) \, dx</math>
+ <math>\int f(x^2) \, dx</math>
- <math>\int F(x) \, dx</math>

{Suppose we know the first three antiderivatives for <math>f</math>, i.e., we have explicit expressions for an antiderivative of <math>f</math>, an antiderivative of that antiderivative, and an antiderivative of the antiderivative of the antiderivative. What is the largest nonnegative integer <math>k</math> for which this guarantees us an expression for an antiderivative of <math>x \mapsto x^kf(x)</math>?
|type="()"}
- 0
- 1
+ 2
- 3
- 4

{Suppose we know the first three antiderivatives for <math>f</math>, i.e., we have explicit expressions for an antiderivative of <math>f</math>, an antiderivative of that antiderivative, and an antiderivative of the antiderivative of the antiderivative. What is the largest nonnegative integer <math>k</math> for which this guarantees us an expression for an antiderivative of <math>x \mapsto f(x^{1/k})</math>? For simplicity, assume that we are only considering <math>x > 0</math>.
|type="()"}
- 0
- 1
- 2
+ 3
- 4
- 5

{Suppose <math>f</math> has a known antiderivative <math>F</math>. Consider the problems of integrating <math>f(x^2), xf(x^2), x^2f(x^2)</math>. What can we say about the relation between these problems?
|type="()"}
- All of these have antiderivatives expressible in terms of <math>F</math>.
- <math>f(x^2)</math> has an antiderivative expressible in terms of <math>F</math>. The integration problems for the other two functions are equivalent to each other.
+ <math>xf(x^2)</math> has an antiderivative expressible in terms of <math>F</math>. The integration problems for the other two functions are equivalent to each other.
- <math>x^2f(x^2)</math> has an antiderivative expressible in terms of <math>F</math>. The integration problems for the other two functions are equivalent to each other.
- All the integration problems are equivalent to each other, but none has a guaranteed expression in terms of <math>f</math> and <math>F</math>.

{Suppose <math>f</math> is an elementarily expressible and infinitely differentiable function on the positive reals (so all derivatives of <math>f</math> are also elementarily expressible). An antiderivative for <math>f''(x)/x</math> is '''not equivalent''' up to elementary functions to '''which one''' of the following?
|type="()"}
- An antiderivative for <math>x \mapsto f''(e^x)</math>, domain all of <math>\R</math>.
+ An antiderivative for <math>x \mapsto f'(e^x/x)</math>, domain positive reals.
- An antiderivative for <math>x \mapsto f'''(x)(\ln x)</math>, domain positive reals.
- An antiderivative for <math>x \mapsto f'(1/x)</math>, domain positive reals.
- An antiderivative for <math>x \mapsto f(1/\sqrt{x})</math>, domain positive reals.

</quiz>

Quiz:Equivalence of integration problems - Revision history

Vipul: /* General functions= */

Vipul: /* Specific functions */

Vipul at 03:43, 20 February 2012

Vipul: Created page with " {Suppose f is a function with a known antiderivative F. Which of the following is correct (and can be deduced using integration..."

Vipul: Created page with " {Suppose $f$ is a function with a known antiderivative $F$ . Which of the following is correct (and can be deduced using integration..."