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Quiz:Equivalence of integration problems: Difference between revisions

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From Calculus

@@ 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]].
+==General functions==
 <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 47: / Line 51: @@
 - 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>
+==Specific functions==
+<quiz display=simple>
+{Suppose <math>a</math> and <math>b</math> are real numbers that are not positive integers. Which of the following is a ''sufficient'' condition for the integration problems <math>\int x^ae^x \, dx</math> and <math>\int x^be^x \, dx</math> to be equivalent?
+|type="()"}
+- <math>a + b</math> is an integer.
++ <math>a - b</math> is an integer.
+|| For simplicity, assume <math>a < b</math> (the process works exactly the same way in reverse if <matH>b < a</math>). Start with the integral <math>\int x^be^x \, dx</math>. Now apply integration by parts taking <math>e^x</math> as the part to integrate and <math>x^b</math> as the part to differentiate. After one application of integration by parts, we need to integrate <math>x^{b-1}e^x</math>. Proceed in the way and we see that we get the integrations of <math>x^be^x, x^{b-1}e^x, x^{b-2}e^x, \dots</math>. If <math>a,b</math> differ by an integer, then after finitely many steps, we will land up with <math>\int x^a e^x\, dx</math>.
+- <math>ab</math> is an integer.
+- <math>a/b</math> is an integer.
+{Suppose <math>a</math> and <math>b</math> are real numbers that are not positive integers. Which of the following is a ''sufficient'' condition for the integration problems <math>\int x^ae^x \, dx</math> and <math>\int e^{x^b} \, dx</math> to be equivalent? Assume we are working with <math>x > 0</math>, so any real power of <math>x</matH> makes sense.
+|type="()"}
+- <math>a + b = 1</math>
+- <math>a - b = 1</math>
++ <math>ab = 1</math>
+|| Using integration by parts once, we can convert <math>\int x^a e^x\, dx</math> to <math>\int ax^{a-1} e^x \, dx</math>. Now, put <math>u = x^a</math>. Then <math>x = u^{1/a}</math>, and <math>du = ax^{a-1} \, dx</math>. So, we get that the integral is <math>\int e^{u^{1/a}} \, du</math>. Replace the dummy variable <math>u</math> by the dummy variable <math>x</math>, to obtain <math>\int e^{x^{1/a}} \, dx</math>, which is <math>\int e^{x^b} \, dx</math> by the assumption that <math>b = 1/a</math>.
+- <math>a/b = 1</math>
+{Suppose <math>a</math> and <math>b</math> are positive real numbers. Which of the following is a ''sufficient'' condition for the integration problems <math>\int e^{x^a} \, dx</math> and <math>\int e^{x^b} \, dx</math> to be equivalent? Assume we are working with <math>x > 0</math>, so any real power of <math>x</matH> makes sense.
+|type="()"}
+- <math>1/a + 1/b</math> is an integer
++ <math>1/a - 1/b</math> is an integer
+|| Put <math>u = x^a</math>. Then,we get <math>x = u^{1/a}</math> and the integral becomes <math>\int e^{x^a} \, dx= \frac{1}{a} \int e^u u^{1/a - 1} \, du</math>. If <matH>1/a - 1/b</math> is an integer, then repeated use of integration by parts gets us to <math>\int e^u u^{1/b - 1} \, du</math>. Now, we plug back <math>y = u^{1/b}</math> and get <math>\int e^{y^b} \, dy</math>. Constants are ignored here as they don't affect the equivalence of integration problems.
+- <matH>1/(ab)</math> is an integer
+- <math>a/b</math> is an integer
+{Which of the following functions has an antiderivative that is '''not equivalent''' up to elementary functions to the antiderivative of <math>x \mapsto e^{-x^2}</math>?
++ <math>x \mapsto e^{-x^4}</math>
+- <math>\mapsto e^{-x^{2/3}}</math>
+|| Equivalent via <math>x \mapsto x^2e^{-x^2}</math>. Start with <math>\int e^{-x^{2/3}}</math>. Do a <math>u</math>-substitution <math>u = x^{1/3}</math>, get <matH>\int 3u^2e^{-u^2} \, du</math>.
+- <math>x \mapsto e^{-x^{2/5}}</math>
+|| Equivalent via <math>x \mapsto x^4e^{-x^2}</math>. Start with <math>\int e^{-x^{2/5}}</math>. Do a <math>u</math>-substitution <math>u = x^{1/5}</math>, get <matH>\int 5u^4e^{-u^2} \, du</math>.
+- <math>x \mapsto x^2e^{-x^2}</math>
+|| Consider <math>\int x^2e^{-x^2} \, dx</math>. Perform integration by parts on this, taking <math>xe^{-x^2} \, dx</math> as the part to integrate.
+- <math>x \mapsto x^4e^{-x^2}</math>
+|| Equivalent via <math>\int x^2e^{-x^2} \, dx</math>. Consider <math>\int x^4 e^{-x^2} \, dx</math>. Split as <matH>x^3 (xe^{-x^2})</math> and take <math>xe^{-x^2}</math> as the part to integrate, and in one step we get to <math>\int x^2e^{-x^2}</math>.
 </quiz>

Latest revision as of 22:06, 18 March 2024

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

1 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 $x \mapsto x f (x)$ is equivalent to knowledge of an antiderivative for $F$ .

2 Suppose $f$ is a function with a known antiderivative $F$ . Which of the following integration problems is not equivalent to the others?

	$\int f (\sqrt{x}) d x$
	$\int x f (x) d x$
	$\int f (x^{2}) d x$
	$\int F (x) d x$

3 Suppose we know the first three antiderivatives for $f$ , i.e., we have explicit expressions for an antiderivative of $f$ , an antiderivative of that antiderivative, and an antiderivative of the antiderivative of the antiderivative. What is the largest nonnegative integer $k$ for which this guarantees us an expression for an antiderivative of $x \mapsto x^{k} f (x)$ ?

	0
	1
	2
	3
	4

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

	0
	1
	2
	3
	4
	5

5 Suppose $f$ has a known antiderivative $F$ . Consider the problems of integrating $f (x^{2}), x f (x^{2}), x^{2} f (x^{2})$ . What can we say about the relation between these problems?

	All of these have antiderivatives expressible in terms of $F$ .
	$f (x^{2})$ has an antiderivative expressible in terms of $F$ . The integration problems for the other two functions are equivalent to each other.
	$x f (x^{2})$ has an antiderivative expressible in terms of $F$ . The integration problems for the other two functions are equivalent to each other.
	$x^{2} f (x^{2})$ has an antiderivative expressible in terms of $F$ . 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 $f$ and $F$ .

6 Suppose $f$ is an elementarily expressible and infinitely differentiable function on the positive reals (so all derivatives of $f$ are also elementarily expressible). An antiderivative for $f^{″} (x) / x$ is not equivalent up to elementary functions to which one of the following?

	An antiderivative for $x \mapsto f^{″} (e^{x})$ , domain all of $R$ .
	An antiderivative for $x \mapsto f^{'} (e^{x} / x)$ , domain positive reals.
	An antiderivative for $x \mapsto f^{‴} (x) (\ln x)$ , domain positive reals.
	An antiderivative for $x \mapsto f^{'} (1 / x)$ , domain positive reals.
	An antiderivative for $x \mapsto f (1 / \sqrt{x})$ , domain positive reals.

Specific functions

1 Suppose $a$ and $b$ are real numbers that are not positive integers. Which of the following is a sufficient condition for the integration problems $\int x^{a} e^{x} d x$ and $\int x^{b} e^{x} d x$ to be equivalent?

	$a + b$ is an integer.
	$a - b$ is an integer.
	$a b$ is an integer.
	$a / b$ is an integer.

2 Suppose $a$ and $b$ are real numbers that are not positive integers. Which of the following is a sufficient condition for the integration problems $\int x^{a} e^{x} d x$ and $\int e^{x^{b}} d x$ to be equivalent? Assume we are working with $x > 0$ , so any real power of $x$ makes sense.

	$a + b = 1$
	$a - b = 1$
	$a b = 1$
	$a / b = 1$

3 Suppose $a$ and $b$ are positive real numbers. Which of the following is a sufficient condition for the integration problems $\int e^{x^{a}} d x$ and $\int e^{x^{b}} d x$ to be equivalent? Assume we are working with $x > 0$ , so any real power of $x$ makes sense.

	$1 / a + 1 / b$ is an integer
	$1 / a - 1 / b$ is an integer
	$1 / (a b)$ is an integer
	$a / b$ is an integer

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