Chapter 6

Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{d x}{3 x-2} $$

In Exercises 1 and \(2,\) complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. $$ \lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 2 x} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} $$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{5}{x^{2}-10 x} $$

Use differentiation to match the antiderivative with the correct integral. [Integrals are labeled (a), (b), (c), and (d).] (a) \(\int \frac{x^{2}}{\sqrt{16-x^{2}}} d x\) (b) \(\int \frac{\sqrt{x^{2}+16}}{x} d x\) (c) \(\int \sqrt{7+6 x-x^{2}} d x\) (d) \(\int \frac{x^{2}}{\sqrt{x^{2}-16}} d x\) $$ 4 \ln \left|\frac{\sqrt{x^{2}+16}-4}{x}\right|+\sqrt{x^{2}+16}+C $$

Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{3} \frac{d x}{x^{2}} $$

Use a table of integrals with forms involving $$\sqrt{u^{2} \pm a^{2}}$$ to find $$\int \frac{\sqrt{x^{2}-9}}{3 x} d x$$

In Exercises 1 and \(2,\) complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. $$ \lim _{x \rightarrow \infty} \frac{6 x}{\sqrt{3 x^{2}-2 x}} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 10 & 10^{2} & 10^{3} & 10^{4} & 10^{5} \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} $$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{4 x^{2}+3}{(x-5)^{3}} $$

Use differentiation to match the antiderivative with the correct integral. [Integrals are labeled (a), (b), (c), and (d).] (a) \(\int \frac{x^{2}}{\sqrt{16-x^{2}}} d x\) (b) \(\int \frac{\sqrt{x^{2}+16}}{x} d x\) (c) \(\int \sqrt{7+6 x-x^{2}} d x\) (d) \(\int \frac{x^{2}}{\sqrt{x^{2}-16}} d x\) $$ 8 \ln \left|\sqrt{x^{2}-16}+x\right|+\frac{x \sqrt{x^{2}-16}}{2}+C $$

Use differentiation to match the antiderivative with the correct integral. [Integrals are labeled (a), (b), (c), and \((\mathbf{d}) .]\) (a) \(\int \sin x \tan ^{2} x d x\) (b) \(8 \int \cos ^{4} x d x\) (c) \(\int \sin x \sec ^{2} x d x\) (d) \(\int \tan ^{4} x d x\) $$ y=x-\tan x+\frac{1}{3} \tan ^{3} x $$

Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x $$

In Exercises \(3-6,\) evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hôpital's Rule. \(\lim _{x \rightarrow 3} \frac{2(x-3)}{x^{2}-9}\)

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-3}{x^{3}+10 x} $$

Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{\infty} \ln \left(x^{2}\right) d x $$

In Exercises \(3-6,\) evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hôpital's Rule. \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{2 x}\)

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-1}{x\left(x^{2}+1\right)^{2}} $$

Find the integral. $$ \int \cos ^{3} x \sin x d x $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{4} \frac{1}{\sqrt{x}} d x $$

In Exercises 5 and 6, use a table of integrals with forms involving the trigonometric functions to find the integral. $$ \int \sin ^{4} 2 x d x $$

In Exercises \(3-6,\) evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hôpital's Rule. \(\lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x+1}{3 x^{2}-5}\)

Use partial fractions to find the integral. $$ \int \frac{1}{x^{2}-1} d x $$

Find the indefinite integral using the substitution \(x=5 \sin \theta\) $$ \int \frac{1}{\left(25-x^{2}\right)^{3 / 2}} d x $$

In Exercises \(5-8,\) identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int x e^{2 x} d x $$

Find the integral. $$ \int \cos ^{3} x \sin ^{4} x d x $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{3}^{4} \frac{1}{(x-3)^{3 / 2}} d x $$

Use a table of integrals with forms involving the trigonometric functions to find the integral. $$ \int \frac{\cos ^{3} \sqrt{x}}{\sqrt{x}} d x $$

In Exercises \(3-6,\) evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hôpital's Rule. \(\lim _{x \rightarrow \infty} \frac{2 x+1}{4 x^{2}+x}\)

Use partial fractions to find the integral. $$ \int \frac{x+1}{x^{2}+4 x+3} d x $$

Find the indefinite integral using the substitution \(x=5 \sin \theta\) $$ \int \frac{x^{2}}{\sqrt{25-x^{2}}} d x $$

Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int \ln 3 x d x $$

Find the integral. $$ \int \sin ^{5} 2 x \cos 2 x d x $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{2} \frac{1}{(x-1)^{2}} d x $$

Use a table of integrals with forms involving \(e^{u}\) to find $$\int \frac{1}{1+e^{2 x}} d x$$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{x-2}\)

Use partial fractions to find the integral. $$ \int \frac{5-x}{2 x^{2}+x-1} d x $$

Find the indefinite integral using the substitution \(x=2 \sin \theta\) $$ \int x^{3} \sqrt{x^{2}-4} d x $$

Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int x \sec ^{2} x d x $$

Find the integral. $$ \int \sin ^{3} x d x $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{0} e^{2 x} d x $$

Use a table of integrals with forms involving $$\ln u$$ to find $$\int(\ln x)^{3} d x$$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow 2^{-}} \frac{\sqrt{4-x^{2}}}{x-2}\)

Use partial fractions to find the integral. $$ \int \frac{5 x^{2}-12 x-12}{x^{3}-4 x} d x $$

Find the indefinite integral using the substitution \(x=2 \sin \theta\) $$ \int \frac{x^{3}}{\sqrt{x^{2}-4}} d x $$

Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int x^{2} \cos x d x $$

Find the integral. $$ \int \sin ^{5} x \cos ^{2} x d x $$

Explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral. Determine whether the utility gives the correct answer. $$ \int_{1}^{1} \frac{1}{x^{2}} d x=-2 $$

In Exercises 9-22, use integration tables to find the integral. $$ \int x \operatorname{arcsec}\left(x^{2}+1\right) d x $$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow 0} \frac{e^{x}-(1-x)}{x}\)

Find the indefinite integral using the substitution \(x=\tan \theta\) $$ \int x \sqrt{1+x^{2}} d x $$

Use partial fractions to find the integral. $$ \int \frac{x^{2}+12 x+12}{x^{3}-4 x} d x $$