Quiz:Equivalence of integration problems

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 $x\mapsto xf(x^{2})$ 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 $x\mapsto xf(x)$ 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?

	$\int f({\sqrt {x}})\,dx$
	$\int xf(x)\,dx$
	$\int f(x^{2})\,dx$
	$\int F(x)\,dx$

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 $f(x^{2}),xf(x^{2}),x^{2}f(x^{2})$. 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.
	$xf(x^{2})$ 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 $\mathbb {R}$.
	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 $\int x^{a}e^{x}\,dx$ and $\int x^{b}e^{x}\,dx$ to be equivalent?

	$a+b$ is an integer.
	$a-b$ is an integer.
	$ab$ 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 $\int x^{a}e^{x}\,dx$ and $\int e^{x^{b}}\,dx$ to be equivalent? Assume we are working with $x>0$, so any real power of $x$ makes sense.

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

3 Suppose $a$ and $b$ are positive real numbers. Which of the following is a sufficient condition for the integration problems $\int e^{x^{a}}\,dx$ and $\int e^{x^{b}}\,dx$ 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
	$1/(ab)$ is an integer
	$a/b$ is an integer