Quiz:Equivalence of integration problems

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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)?

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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.

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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