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

 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 $x > 0$.

 1 2 3 4 5

5 Suppose $f$ has a known antiderivative $F$. Consider the problems of integrating $f(x^2), xf(x^2), x^2f(x^2)$. 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. $x^2f(x^2)$ 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 $\int x^ae^x \, dx$ and $\int x^be^x \, dx$ 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 $\int x^ae^x \, dx$ 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