Chapter 7

What is the order of \(y^{\prime \prime}(t)+9 y(t)=10 ?\)

What are the two general ways in which an improper integral may occur?

If the interval [4,18] is partitioned into \(n=28\) sub-intervals of equal length, what is \(\Delta x ?\)

Give some examples of analytical methods for evaluating integrals.

What change of variables is suggested by an integral containing \(\sqrt{x^{2}-9} ?\)

What kinds of functions can be integrated using partial fraction decomposition?

State the half-angle identities used to integrate \(\sin ^{2} x\) and \(\cos ^{2} x\).

On which derivative rule is integration by parts based?

What change of variables would you use for the integral \(\int(4-7 x)^{-6} d x ?\)

Is \(y^{\prime \prime}(t)+9 y(t)=10\) linear or nonlinear?

Explain geometrically how the Midpoint Rule is used to approximate a definite integral.

Does a computer algebra system give an exact result for an indefinite integral? Explain.

What change of variables is suggested by an integral containing \(\sqrt{x^{2}+36} ?\)

Give an example of each of the following. a. A simple linear factor b. A repeated linear factor c. A simple irreducible quadratic factor d. A repeated irreducible quadratic factor

State the three Pythagorean identities.

How would you choose \(d v\) when evaluating \(\int x^{n} e^{a x} d x\) using integration by parts?

Before integrating, how would you rewrite the integrand of \(\int\left(x^{4}+2\right)^{2} d x ?\)

Explain how to evaluate \(\int_{0}^{1} x^{-1 / 2} d x\)

Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.

Why might an integral found in a table differ from the same integral evaluated by a computer algebra system?

What change of variables is suggested by an integral containing \(\sqrt{100-x^{2}} ?\)

What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following? a. A factor of \(x-3\) in the denominator b. A factor of \((x-4)^{3}\) in the denominator c. A factor of \(x^{2}+2 x+6\) in the denominator

Describe the method used to integrate \(\sin ^{3} x\).

What trigonometric identity is useful in evaluating \(\int \sin ^{2} x d x ?\)

If the general solution of a differential equation is \(y=c e^{-3 t}+10,\) what is the solution that satisfies the initial condition \(y(0)=5 ?\)

For what values of \(p\) does \(\int_{1}^{\infty} x^{-p} d x\) converge?

If the Midpoint Rule is used on the interval [-1,11] with \(n=3\) sub-intervals, at what \(x\) -coordinates is the integrand evaluated?

Is a reduction formula an analytical method or a numerical method? Explain.

If \(x=4 \tan \theta,\) express \(\sin \theta\) in terms of \(x\)

What is the first step in integrating \(\frac{x^{2}+2 x-3}{x+1} ?\)

Describe the method used to integrate \(\sin ^{m} x \cos ^{n} x,\) for \(m\) even and \(n\) odd.

Describe a first step in integrating \(\int \frac{x^{3}-2 x+4}{x-1} d x.\)

What is a separable first-order differential equation?

Evaluate the following integrals or state that they diverge. $$\int_{1}^{\infty} x^{-2} d x$$

If the Trapezoid Rule is used on the interval [-1,9] with \(n=5\) sub-intervals, at what \(x\) -coordinates is the integrand evaluated?

If \(x=2 \sin \theta,\) express \(\cot \theta\) in terms of \(x\)

Give the partial fraction decomposition for the following functions. $$\frac{2}{x^{2}-2 x-8}$$

What type of integrand is a good candidate for integration by parts?

Is the equation \(t^{2} y^{\prime}(t)=(t+4) / y^{2}\) separable?

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\infty} \frac{d x}{(x+1)^{3}}$$

State how to compute the Simpson's Rule approximation \(S(2 n)\) if the Trapezoid Rule approximations \(T(2 n)\) and \(T(n)\) are known.

Use a table of integrals to determine the following indefinite integrals. $$\int \sin 3 x \cos 2 x d x$$

If \(x=8 \sec \theta,\) express tan \(\theta\) in terms of \(x\)

Give the partial fraction decomposition for the following functions. $$\frac{x-9}{x^{2}-3 x-18}$$

How would you evaluate \(\int \cos ^{2} x \sin ^{3} x d x ?\)

Describe a first step in integrating \(\int \frac{x^{10}-2 x^{4}+10 x^{2}+1}{3 x^{3}} d x.\)

Explain how to solve a separable differential equation of the form \(g(y) y^{\prime}(t)=h(t)\).

Evaluate the following integrals or state that they diverge. $$\int_{-\infty}^{0} e^{x} d x$$

Compute the absolute and relative errors in using c to approximate \(x\). \(x=\pi ; c=3.14\)

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d x}{\sqrt{x^{2}+16}}$$