Problem 77
Determine whether the following integrals converge or diverge. $$\int_{1}^{\infty} \frac{d x}{x^{3}+1}$$
Problem 77
Trapezoid Rule and concavity Suppose \(f\) is positive and its first two derivatives are continuous on \([a, b] .\) If \(f^{\prime \prime}\) is positive on \([a, b]\) then is a Trapezoid Rule estimate of \(\int_{a}^{b} f(x) d x\) an underestimate or overestimate of the integral? Justify your answer using Theorem 8.1 and an illustration.
Problem 77
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.
Problem 77
Evaluate the following integrals. $$\int \cos ^{-1} x d x$$
Problem 77
Tabular integration Consider the integral \(\int f(x) g(x) d x,\) where \(f\) can be differentiated repeatedly and \(g\) can be integrated repeatedly. Let \(G_{k}\) represent the result of calculating \(k\) indefinite integrals of \(g,\) where the constants of integration are omitted. a. Show that integration by parts, when applied to \(\int f(x) g(x) d x\) with the choices \(u=f(x)\) and \(d v=g(x) d x,\) leads to \(\int f(x) g(x) d x=f(x) G_{1}(x)-\int f^{\prime}(x) G_{1}(x) d x .\) This formula can be remembered by utilizing the following table, where a right arrow represents a product of functions on the right side of the integration by parts formula, and a left arrow represents the integral of a product of functions (also appearing on the right side of the formula). Explain the significance of the signs associated with the arrows. (Table cannot copy) b. Perform integration by parts again on \(\int f^{\prime}(x) G_{1}(x) d x\) (from part (a)) with the choices \(u=f^{\prime}(x)\) and \(d v=G_{1}(x) d x\) to show that \(\int f(x) g(x) d x=f(x) G_{1}(x)-f^{\prime}(x) G_{2}(x)+\) \(\int f^{\prime \prime}(x) G_{2}(x) d x .\) Explain the connection between this integral formula and the following table, paying close attention to the signs attached to the arrows. (Table cannot copy) c. Continue the pattern established in parts (a) and (b) and integrate by parts a third time. Write the integral formula that results from three applications of integration by parts, and construct the associated tabular integration table (include signs of the arrows). d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts. Evaluate \(\int x^{2} e^{x / 2} d x\) by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in. e. Use tabular integration to evaluate \(\int x^{3} \cos x d x .\) How many rows of the table are necessary? Why? f. Explain why tabular integration is particularly suited to integrals of the form \(\int p_{n}(x) g(x) d x,\) where \(p_{n}\) is a polynomial of degree \(n>0\) (and where, as before, we assume \(g\) is easily integrated as many times as necessary).
Problem 78
Surface area Find the area of the surface generated when the region bounded by the graph of \(y=e^{x}+\frac{1}{4} e^{-x}\) and the \(x\) -axis on the interval \([0, \ln 2]\) is revolved about the \(x\) -axis.
Problem 78
Shortcut for Simpson's Rule Using the notation of the text, prove that \(S(2 n)=\frac{4 T(2 n)-T(n)}{3},\) for \(n \geq 1\)
Problem 78
On the interval [0,2] , the graphs of \(f(x)=x^{2} / 3\) and \(g(x)=x^{2}\left(9-x^{2}\right)^{-1 / 2}\) have similar shapes. a. Find the area of the region bounded by the graph of \(f\) and the \(x\) -axis on the interval [0,2] b. Find the area of the region bounded by the graph of \(g\) and the \(x\) -axis on the interval [0,2] c. Which region has greater area?
Problem 78
Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77 ). a. \(\int x^{4} e^{x} d x \quad\) b. \(\int 7 x e^{3 x} d x\) c. \(\int_{-1}^{0} 2 x^{2} \sqrt{x+1} d x\) d. \(\int\left(x^{3}-2 x\right) \sin 2 x \, d x\) e. \(\int \frac{2 x^{2}-3 x}{(x-1)^{3}} d x\) f. \(\int \frac{x^{2}+3 x+4}{\sqrt[3]{2 x+1}} d x\) g. Why doesn't tabular integration work well when applied to \(\int \frac{x}{\sqrt{1-x^{2}}} d x \, ?\) Evaluate this integral using a different method.
Problem 78
Determine whether the following integrals converge or diverge. $$\int_{0}^{\infty} \frac{d x}{e^{x}+x+1}$$
