Quiz: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 $F^{\prime }G$, we can obtain an expression for the antiderivative for the pointwise product $F{G}^{\prime }$.
	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?

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

Choosing the parts to integrate and differentiate

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 $x\mapsto x^{2}f(x)$ is equivalent to knowledge of an antiderivative for $F$.