Definite integrals Use a computer algebra system to

Give some examples of analytical methods for evaluating integrals.

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

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

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{\sqrt{x^{2}+16}}\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{3 u}{2 u+7} d u\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{\sqrt{x^{2}-25}}\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d y}{y(2 y+9)}\)

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

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{1-\cos 4 x}\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{x \sqrt{81-x^{2}}}\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \sqrt{4 x+12} d x\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{\sqrt{9 x^{2}-100}}\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{\sqrt{4 x+1}}\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{225-16 x^{2}}\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{\left(16+9 x^{2}\right)^{3 / 2}}\)

Arc length of a parabola Let L(c) be the length of the parabola \(f(x)=x^{2}\) from x=0 to x=c, where \(c \geq 0\) is a constant. (a) Find an expression for L and graph the function. (b) Is L concave up or concave down on \([0, \infty)\)? (c) Show that as c becomes large and positive, the arc length function increases as \(c^{2}\); that is, \(L(c) \approx k c^{2}\), where k is a constant.

Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is an integer. \(\int x^{n} \sin ^{-1} x \ d x\); Use integration by parts.

Geometry problems Use a table of integrals to solve the following problems. Find the length of the curve \(y=x^{2} / 4\) on the interval [0,8].

Geometry problems Use a table of integrals to solve the following problems. The region bounded by the graphs of \(y=\pi / 2\), \(y=\sin ^{-1} x\), and the y-axis is revolved about the y-axis. What is the volume of the solid that is formed?

Geometry problems Use a table of integrals to solve the following problems. Find the length of the curve \(y=e^{x}\) on the interval [0, ln 2].

Geometry problems Use a table of integrals to solve the following problems. The region bounded by the graph of \(y=\frac{1}{\sqrt{x+4}}\) and the x-axis on the interval [0,12] is revolved about the y-axis. What is the volume of the solid that is formed?

Indefinite integrals Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. \(\int\left(a^{2}-x^{2}\right)^{3 / 2} d x\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{\cos ^{-1} \sqrt{x}}{\sqrt{x}} d x\)

Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals. \(\int x^{2} e^{-3 x} d x\)

Indefinite integrals Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. \(\int \sqrt{4 x^{2}+36} d x\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \sqrt{4 x^{2}-9} d x\)

Double table lookup The following integrals may require more than one table lookup. Evaluate the integrals using a table of integrals, then check your answer with a computer algebra system. When the parameter a appears, assume a>0. \(\int \frac{d x}{\sqrt{2 a x-x^{2}}}\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{\tan ^{-1} x^{3}}{x^{4}} d x\)

Geometry problems Use a table of integrals to solve the following problems. The region bounded by the graph of y=1/(x+10) and the x-axis on the interval [0,3] is revolved about the x-axis. What is the volume of the solid that is formed?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. (a) It is possible that a computer algebra system says \(\int \frac{d x}{x(x-1)}=\ln (x-1)-\ln x\) and a table of integrals says \(\int \frac{d x}{x(x-1)}=\ln \left|\frac{x-1}{x}\right|+C\). (b) A computer algebra system working in symbolic mode could give the result \(\int_{0}^{1} x^{8} d x=\frac{1}{9}\), and a computer algebra system working in approximate (numerical) mode could give the result \(\int_{0}^{1} x^{8} d x=0.11111111\).

Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is an integer. \(\int x(a x+b)^{n} d x\); Use u=ax+b.

Definite integrals Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. \(\int_{0}^{2 \pi} \frac{d x}{(4+2 \sin x)^{2}}\)

Apparent discrepancy Resolve the apparent discrepancy between \(\int \frac{d x}{x(x-1)(x+2)}=\frac{1}{6} \ln \frac{(x-1)^{2}|x+2|}{|x|^{3}}+C\) and \(\int \frac{d x}{x(x-1)(x+2)}=\frac{\ln |x-1|}{3}+\frac{\ln |x+2|}{6}-\frac{\ln |x|}{2}+C\)

Indefinite integrals Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. \(\int \frac{\left(x^{2}-a^{2}\right)^{3 / 2}}{x} d x\)

Definite integrals Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. \(\int_{0}^{\pi / 4} \ln (1+\tan x) d x\)

Indefinite integrals Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. \(\int \frac{d x}{x\left(a^{2}-x^{2}\right)^{2}}\)

Indefinite integrals Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. \(\int \tan ^{2} 3 x d x\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{x\left(x^{8}-256\right)}\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{\cos x}{\sin ^{2} x+2 \sin x} d x\)

Apparent discrepancy Three different computer algebra systems give the following results: \(\int \frac{d x}{x \sqrt{x^{4}-1}}=\frac{1}{2} \cos ^{-1} \sqrt{x^{-4}}=\frac{1}{2} \cos ^{-1} x^{-2}=\frac{1}{2} \tan ^{-1} \sqrt{x^{4}-1}\) Explain how they can all be correct.

Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is an integer. \(\int \frac{x}{a x+b} d x\); Use u=ax+b.

Definite integrals Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. \(\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{x \sqrt{144-x^{2}}}\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{\sqrt{\ln ^{2} x+4}}{x} d x\)

Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is an integer. \(\int \frac{x}{\sqrt{a x+b}} d x\); Use \(u^{2}\) = ax + b.

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \sqrt{x^{2}-4 x+8} d x\)

Definite integrals Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. \(\int_{0}^{1} \ln x \ln (1+x) d x\)

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{x\left(x^{3}+8\right)}\)

Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals. \(\int \tan ^{4} 3 y d y\)

Double table lookup The following integrals may require more than one table lookup. Evaluate the integrals using a table of integrals, then check your answer with a computer algebra system. When the parameter a appears, assume a>0. \(\int x \sin ^{-1} 2 x d x\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{d x}{\sqrt{x^{2}-6 x}}\)

Geometry problems Use a table of integrals to solve the following problems. Find the area of the region bounded by the graph of \(y=\frac{1}{\sqrt{x^{2}-2 x+2}}\) and the x-axis between x=0 and x=3.

Double table lookup The following integrals may require more than one table lookup. Evaluate the integrals using a table of integrals, then check your answer with a computer algebra system. When the parameter a appears, assume a>0. \(\int \sqrt{2 a x-x^{2}} d x\)

Geometry problems Use a table of integrals to solve the following problems. Find the length of the curve \(y=x^{3 / 2}+8\) on the interval [0,2].

Table lookup integrals Use a table of integrals to evaluate the following indefinite integrals. \(\int \frac{d x}{x\left(x^{10}+1\right)}\)

Definite integrals Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. \(\int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{2} x}\)

Geometry problems Use a table of integrals to solve the following problems. The graphs of \(f(x)=\frac{2}{x^{2}+1}\) and \(g(x)=\frac{7}{4 \sqrt{x^{2}+1}}\) are shown in the accompanying figure. Which is greater, the average value of f or that of g on the interval [-1,1]?

Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals. \(\int x^{3} e^{2 x} d x\)

Indefinite integrals Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. \(\int \frac{x}{\sqrt{2 x+3}} d x\)

Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals. \(\int \sec ^{4} 4 x d x\)

A remarkable integral It is a fact that \(\int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{m} x}=\frac{\pi}{4}\) for all real numbers m. (a) Graph the integrand for m=-2, -3/2, -1, -1/2, 0, 1/2, 1, 3/2, and 2 , and explain geometrically how the area under the curve on the interval \([0, \pi / 2]\) remains constant as m varies. (b) Use a computer algebra system to confirm that the integral is constant for all m.

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{d x}{\sqrt{x^{2}+10 x}}\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{d x}{x^{2}+2 x+10}\)

Indefinite integrals Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. \(\int\left(a^{2}-x^{2}\right)^{-2} d x\)

Definite integrals Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. \(\int_{1 / 2}^{1} \frac{\sin ^{-1} x}{x} d x\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{e^{x}}{\sqrt{e^{2 x}+4}} d x\)

Double table lookup The following integrals may require more than one table lookup. Evaluate the integrals using a table of integrals, then check your answer with a computer algebra system. When the parameter a appears, assume a>0. \(\int \frac{\tan ^{-1} x}{x^{2}} d x\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{\ln x \sin ^{-1}(\ln x)}{x} d x\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{d t}{\sqrt{1+4 e^{t}}}\)

Double table lookup The following integrals may require more than one table lookup. Evaluate the integrals using a table of integrals, then check your answer with a computer algebra system. When the parameter a appears, assume a>0. \(\int \frac{\sin ^{-1} a x}{x^{2}} d x\)

Definite integrals Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. \(\int_{2 / 3}^{4 / 5} x^{8} d x\)

Definite integrals Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results. \(\int_{0}^{\pi / 2} \cos ^{6} x d x\)

Double table lookup The following integrals may require more than one table lookup. Evaluate the integrals using a table of integrals, then check your answer with a computer algebra system. When the parameter a appears, assume a>0. \(\int 4 x \cos ^{-1} 10 x \ d x\)

Indefinite integrals Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. \(\int\left(x^{2}+a^{2}\right)^{-5 / 2} d x\)

Preliminary work Use a table of integrals to evaluate the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. \(\int \frac{e^{t}}{\sqrt{3+4 e^{t}}} d t\)

Reconciling results Using one computer algebra system, it was found that \(\int \frac{d x}{1+\sin x}=\frac{\sin x-1}{\cos x}\) and using another computer algebra system, it was found that \(\int \frac{d x}{1+\sin x}=\frac{2 \sin (x / 2)}{\cos (x / 2)+\sin (x / 2)}\). Reconcile the two answers.

Period of a pendulum Consider a pendulum with a length of L meters swinging only under the influence of gravity. Suppose the pendulum starts swinging with an initial displacement of \(\theta_{0}\) radians (see figure). The period (time to complete one full cycle) is given by \(T=\frac{4}{\omega} \int_{0}^{\pi / 2} \frac{d \varphi}{\sqrt{1-k^{2} \sin ^{2} \varphi}}\) where \(\omega^{2}=g / L\), \(g \approx 9.8 \mathrm{~m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(k^{2}=\sin ^{2}\left(\theta_{0} / 2\right)\). Assume L=9.8 m, which means \(\omega=1 \mathrm{~s}^{-1}\). (a) Use a computer algebra system to find the period of the pendulum for \(\theta_{0}=0.1,0.2\), \(\ldots\), 0.9,1.0 rad. (b) For small values of \(\theta_{0}\), the period should be approximately \(2 \pi \mathrm{s}\). For what values of \(\theta_{0}\) are your computed values within 10 % of \(2 \pi\) (relative error less than 0.1)?

Powers of sine and cosine It can be shown that \(\int_{0}^{\pi / 2} \sin ^{n} x \ d x=\int_{0}^{\pi / 2} \cos ^{n} x \ d x=\) \(\begin{cases}\frac{1 \cdot 3 \cdot 5 \cdots(n-1)}{2 \cdot 4 \cdot 6 \cdots n} \cdot \frac{\pi}{2} & \text { if } n \geq 2 \text { is an even integer } \\ \frac{2 \cdot 4 \cdot 6 \cdots(n-1)}{3 \cdot 5 \cdot 7 \cdots n} & \text { if } n \geq 3 \text { is an odd integer }\end{cases}\) (a) Use a computer algebra system to confirm this result for n=2,3,4, and 5 . (b) Evaluate the integrals with n=10 and confirm the result. (c) Using graphing and/or symbolic computation, determine whether the values of the integrals increase or decrease as n increases.