Chapter 5

Evaluate the integral. \(\int_{4}^{9} \frac{\ln y}{\sqrt{y}} d y\)

\(15-17\) Use Definition 2 to find an expression for the area under the graph of \(f\) as a limit. Do not evaluate the limit. $$f(x)=x \cos x, \quad 0 \leqq x \leqslant \pi / 2$$

Make a substitution to express the integrand as a rational function and then evaluate the integral. $$\int \frac{e^{2 x}}{e^{2 x}+3 e^{x}+2} d x$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{1}^{\infty} \frac{\ln x}{x} d x\)

Evaluate the integral. \(\int_{1}^{9} \frac{1}{2 x} d x\)

Evaluate the indefinite integral. \(\int \frac{a+b x^{2}}{\sqrt{3 a x+b x^{3}}} d x\)

Evaluate the integral. \(\int_{0}^{1} \frac{y}{e^{2 y}} d y\)

$$\begin{array}{c}{\text { 18. (a) Use Definition } 2 \text { to find an expression for the area under }} \\ {\text { the curve } y=x^{3} \text { from } 0 \text { to } 1 \text { as a limit. }} \\ {\text { (b) The following formula for the sum of the cubes of the }} \\ {\text { first } n \text { integers is proved in Appendix E. Use it to evalu- }} \\ {\text { ate the limit in part (a). }} \\ {\quad 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}}\end{array}$$

Make a substitution to express the integrand as a rational function and then evaluate the integral. $$\int \frac{\cos x}{\sin ^{2} x+\sin x} d x$$

Use the Table of Integrals on Reference Pages \(6-10\) to evaluate the integral. \(\int e^{t} \sin (\alpha t-3) d t\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{-\infty}^{\infty} x^{3} e^{-x^{4}} d x\)

Evaluate the integral. \(\int_{0}^{5}\left(2 e^{x}+4 \cos x\right) d x\)

Evaluate the indefinite integral. \(\int \frac{z^{2}}{z^{3}+1} d z\)

Evaluate the integral. \(\int_{1}^{\sqrt{3}} \arctan (1 / x) d x\)

$$\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[4-3\left(x_{i}^{*}\right)^{2}+6\left(x_{i}^{*}\right)^{5}\right] \Delta x, \quad[0,2]$$

If a linear factor in the denominator of a rational function is repeated, there will be two corresponding partial fractions. For instance, $$f(x)=\frac{5 x^{2}+3 x-2}{x^{2}(x+2)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x+2}$$ $$\begin{array}{l}{\text { Determine the values of } A, B, \text { and } C \text { and use them to }} \\ {\text { evaluate } \int f(x) d x .}\end{array}$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{-\infty}^{\infty} \frac{x^{2}}{9+x^{6}} d x\)

Evaluate the integral. \(\int_{0}^{1}\left(x^{e}+e^{x}\right) d x\)

Evaluate the indefinite integral. \(\int e^{x} \sqrt{1+e^{x}} d x\)

Evaluate the integral. \(\int_{1}^{2}(\ln x)^{2} d x\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{1}^{\infty} \frac{\ln x}{x^{3}} d x\)

Evaluate the integral. \(\int_{0}^{1} 10^{x} d x\)

Evaluate the indefinite integral. \(\int \sec 2 \theta \tan 2 \theta d \theta\)

Evaluate the integral. \(\int_{0}^{1} \frac{r^{3}}{\sqrt{4+r^{2}}} d r\)

21\. (a) Express the area under the curve \(y=x^{3}\) from 0 to 2 as a limit. (b) Use a computer algebra system to find the sum in your expression from part (a). (c) Evaluate the limit in part (a).

$$\begin{array}{c}{\text { If a factor of the denominator is an irreducible quadratic, }} \\ {\text { such as } x^{2}+1, \text { the corresponding partial fraction has a }} \\ {\text { linear numerator. For instance, }} \\\ {f(x)=\frac{2 x^{2}+x+1}{x\left(x^{2}+1\right)}=\frac{A}{x}+\frac{B x+C}{x^{2}+1}}\end{array}$$ $$\begin{array}{l}{\text { Determine the values of } A, B \text { , and } C \text { and use them to }} \\ {\text { evaluate } f f(x) d x .}\end{array}$$

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \sec ^{4} x d x\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{e}^{\infty} \frac{1}{x(\ln x)^{3}} d x\)

Evaluate the integral. \(\int_{-1}^{1} e^{u+1} d u\)

Evaluate the indefinite integral. \(\int \frac{\cos x}{\sin ^{2} x} d x\)

First make a substitution and then use integration by parts to evaluate the integral. \(\int \cos \sqrt{x} d x\)

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int x^{2}\left(1+x^{3}\right)^{4} d x\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{0}^{\infty} \frac{e^{x}}{e^{2 x}+3} d x\)

Evaluate the integral. \(\int_{0}^{1} \frac{4}{t^{2}+1} d t\)

Evaluate the indefinite integral. \(\int \frac{\tan ^{-1} x}{1+x^{2}} d x\)

First make a substitution and then use integration by parts to evaluate the integral. \(\int t^{3} e^{-t^{2}} d t\)

23\. Find the exact area under the cosine curve \(y=\cos x\) from \(x=0\) to \(x=b,\) where 0\(\leqslant b \leqslant \pi / 2 .\) (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if \(b=\pi / 2 ?\)

$$\begin{array}{c}{\text { Suppose that } F, G, \text { and } Q \text { are polynomials and }} \\ {\frac{F(x)}{Q(x)}=\frac{G(x)}{Q(x)}}\end{array}$$ $$\begin{array}{l}{\text { for all } x \text { except when } Q(x)=0 . \text { Prove that } F(x)=G(x) \text { for }} \\ {\text { all } x .[\text {Hint} \text { : Use continuity.] }}\end{array}$$

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int x \sqrt{1+2 x} d x\)

Sketch the region and find its area (if the area is finite). \(S=\\{(x, y) | x \leqslant 1,0 \leqslant y \leqslant e^{x}\\}\)

Evaluate the integral. \(\int_{1}^{2} \frac{v^{3}+3 v^{6}}{v^{4}} d v\)

Evaluate the indefinite integral. \(\int\left(x^{2}+1\right)\left(x^{3}+3 x\right)^{4} d x\)

First make a substitution and then use integration by parts to evaluate the integral. \(\int_{\sqrt{\pi / 2}}^{\sqrt{\pi}} \theta^{3} \cos \left(\theta^{2}\right) d \theta\)

$$\begin{array}{l}{\text { Sterile insect technique One method of slowing the }} \\ {\text { growth of an insect population without using pesticides is }} \\\ {\text { to introduce into the population a number of sterile males }} \\\ {\text { that mate with fertile females but produce no offspring. }}\end{array}$$. $$ \begin{array}{l}{\text { (The photo shows a screw-worm fly, the first pest effec- }} \\ {\text { tively eliminated from a region by this method.) Let } P} \\ {\text { represent the number of female insects in a population and }} \\\ {S \text { the number of sterile males introduced each generation. }}\end{array}$$. $$\begin{array}{c}{\text { Let } r \text { be the per capita rate of production of females by }} \\ {\text { females, provided their chosen mate is not sterile. Then the }} \\ {\text { female population is related to time } t \text { by }} \\ {t=\int \frac{P+S}{P[(r-1) P-S]} d P}\end{array}$$ $$\begin{array}{l}{\text { Suppose an insect population with } 10,000 \text { females grows at }} \\ {\text { a rate of } r=1.1 \text { and } 900 \text { sterile males are added. Evaluate }} \\ {\text { the integral to give an equation relating the female popula- }} \\ {\text { tion to time. (Note that the resulting equation can't be }} \\ {\text { solved explicitly for } P .}\end{array}$$

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \frac{d x}{e^{x}\left(3 e^{x}+2\right)}\)

Sketch the region and find its area (if the area is finite). \(S=\\{(x, y) | x \geqslant-2,0 \leqslant y \leqslant e^{-x / 2}\\}\)

Evaluate the integral. \(\int_{0}^{\pi / 3} \frac{\sin \theta+\sin \theta \tan ^{2} \theta}{\sec ^{2} \theta} d \theta\)

Evaluate the indefinite integral. \(\int \frac{\sin (\ln x)}{x} d x\)

First make a substitution and then use integration by parts to evaluate the integral. \(\int_{0}^{\pi} e^{\cos t} \sin 2 t d t\)

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \tan ^{5} x d x\)