Quiz:Integration by parts

For background, see integration by parts.

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 $fg$ 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 $\!FG'$.
	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?

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

Equivalence of integration problems

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 integration types

1 Suppose $p$ is a polynomial function. In order to find the indefinite integral for a function of the form $x\mapsto p(x)\sin x$, the general strategy, which always works, is to take $p(x)$ as the part to differentiate and $\sin x$ as the part to integrate, and keep repeating the process. Which of the following is the best explanation for why this strategy works?

	$\sin$ can be repeatedly differentiated and polynomials can be repeatedly integrated, giving polynomials all the way.
	$\sin$ can be repeatedly integrated and polynomials can be repeatedly differentiated, eventually becoming zero.
	$\sin$ and polynomials can both be repeatedly differentiated.
	$\sin$ and polynomials can both be repeatedly integrated.

2 Suppose $p$ is a polynomial function. In order to find the indefinite integral for a function of the form $x\mapsto p(x)\exp(x)$, the general strategy, which always works, is to take $p(x)$ as the part to differentiate and $\exp(x)$ as the part to integrate, and keep repeating the process. Which of the following is the best explanation for why this strategy works?

	$\exp$ can be repeatedly differentiated and polynomials can be repeatedly integrated, giving polynomials all the way.
	$\exp$ can be repeatedly integrated and polynomials can be repeatedly differentiated, eventually becoming zero.
	$\exp$ and polynomials can both be repeatedly differentiated.
	$\exp$ and polynomials can both be repeatedly integrated.

3 Consider the function $x\mapsto \exp(x)\sin x$. This function can be integrated using integration by parts. What can we say about how integration by parts works?

	We choose $\exp$ as the part to integrate and $\sin$ as the part to differentiate, and apply this process once to get the answer directly.
	We choose $\exp$ as the part to integrate and $\sin$ as the part to differentiate, and apply this process once, then use a recursive method (identify the integrals on the left and right side) to get the answer.
	We choose $\exp$ as the part to integrate and $\sin$ as the part to differentiate, and apply this process twicce to get the answer directly.
	We choose $\exp$ as the part to integrate and $\sin$ as the part to differentiate, and apply this process twice, then use a recursive method (identify the integrals on the left and right side) to get the answer.

4 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.

5 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.

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

6 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