Chapter 8

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{d x}{5 x-3} $$

Numerical and Graphical Analysis In Exercises \(1-4\) , complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. $$ \lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x} $$ $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0.001} & {0.01} & {0.1} \\ \hline f(x) & {} & {} & {} \\ \hline\end{array} $$

Partial Fraction Decomposition In Exercises \(1-4,\) write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{4}{x^{2}-8 x} $$

Trigonometric Substitution In Exercises \(1-4,\) state the trigonometric substitution you would use to find the indefinite integral. Do not integrate. $$ \int\left(9+x^{2}\right)^{-2} d x $$

Finding an Indefinite Integral Involving Sine and cosine In Exercises \(1-12,\) find the indefinite integral. $$ \int \cos ^{5} x \sin x d x $$

Choosing an Antiderivative In Exercises \(1-4,\) select the correct antiderivative. \(\frac{d y}{d x}=\frac{x}{\sqrt{x^{2}+1}}\) $$ \begin{array}{ll}{\text { (a) } 2 \sqrt{x^{2}+1}+C} & {\text { (b) } \sqrt{x^{2}+1}+C} \\ {\text { (c) } \frac{1}{2} \sqrt{x^{2}+1}+C} & {\text { (d) } \ln \left(x^{2}+1\right)+C}\end{array} $$

In Exercises 1–6, identify and for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int x e^{2 x} d x $$

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{2} \frac{d x}{x^{3}} $$

Numerical and Graphical Analysis In Exercises \(1-4\) , complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. $$ \lim _{x \rightarrow 0} \frac{1-e^{x}}{x} $$ $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0.001} & {0.01} & {0.1} \\ \hline f(x) & {} & {} & {} \\ \hline\end{array} $$

Partial Fraction Decomposition In Exercises \(1-4,\) write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x^{2}+1}{(x-3)^{3}} $$

Use a table of integrals with forms involving \(a+b u\) to find the indefinite integral. \(\int \frac{2}{x^{2}(4+3 x)^{2}} d x\)

Trigonometric Substitution In Exercises \(1-4,\) state the trigonometric substitution you would use to find the indefinite integral. Do not integrate. $$ \int \sqrt{4-x^{2}} d x $$

Finding an Indefinite Integral Involving Sine and cosine In Exercises \(1-12,\) find the indefinite integral. $$ \int \cos ^{3} x \sin ^{4} x d x $$

Choosing an Antiderivative In Exercises \(1-4,\) select the correct antiderivative. \(\frac{d y}{d x}=\frac{x}{x^{2}+1}\) $$ \begin{array}{ll}{\text { (a) } \ln \sqrt{x^{2}+1}+c} & {\text { (b) } \frac{2 x}{\left(x^{2}+1\right)^{2}}+c} \\ {\text { (c) arctan } x+c} & {\text { (d) } \ln \left(x^{2}+1\right)+c}\end{array} $$

In Exercises 1–6, identify and for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int x^{2} e^{2 x} d x $$

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x $$

Numerical and Graphical Analysis In Exercises \(1-4\) , complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. $$ \lim _{x \rightarrow \infty} x^{5} e^{-x / 100} $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {10} & {10^{2}} & {10^{3}} & {10^{4}} & {10^{5}} \\ \hline f(x) & {} & {} & {} & {} \\ \hline\end{array} $$

Partial Fraction Decomposition In Exercises \(1-4,\) write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-3}{x^{3}+10 x} $$

Use a table of integrals with forms involving \(\sqrt{a^{2}-u^{2}}\) to find the indefinite integral. \(\int \frac{1}{x^{2} \sqrt{1-x^{2}}} d x\)

Trigonometric Substitution In Exercises \(1-4,\) state the trigonometric substitution you would use to find the indefinite integral. Do not integrate. $$ \int \frac{x^{2}}{\sqrt{25-x^{2}}} d x $$

Finding an Indefinite Integral Involving Sine and cosine In Exercises \(1-12,\) find the indefinite integral. $$ \int \sin ^{7} 2 x \cos 2 x d x $$

Choosing an Antiderivative In Exercises \(1-4,\) select the correct antiderivative. \(\frac{d y}{d x}=\frac{1}{x^{2}+1}\) $$ \begin{array}{ll}{\text { (a) } \ln \sqrt{x^{2}+1}+C} & {\text { (b) } \frac{2 x}{\left(x^{2}+1\right)^{2}}+C} \\ {\text { (c) } \arctan x+c} & {\text { (d) } \ln \left(x^{2}+1\right)+c}\end{array} $$

In Exercises 1–6, identify and for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int(\ln x)^{2} d x $$

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{\infty} \ln \left(x^{2}\right) d x $$

Partial Fraction Decomposition In Exercises \(1-4,\) write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-1}{x\left(x^{2}+1\right)^{2}} $$

Use a table of integrals with forms involving \(\sqrt{a^{2}-u^{2}}\) to find the indefinite integral. \(\int \frac{\sqrt{64-x^{4}}}{x} d x\)

Trigonometric Substitution In Exercises \(1-4,\) state the trigonometric substitution you would use to find the indefinite integral. Do not integrate. $$ \int x^{2}\left(x^{2}-25\right)^{3 / 2} d x $$

Finding an Indefinite Integral Involving Sine and cosine In Exercises \(1-12,\) find the indefinite integral. $$ \int \sin ^{3} 3 x d x $$

Choosing an Antiderivative In Exercises \(1-4,\) select the correct antiderivative. \(\frac{d y}{d x}=x \cos \left(x^{2}+1\right)\) $$ \begin{array}{ll}{\text { (a) } 2 x \sin \left(x^{2}+1\right)+C} & {\text { (b) }-\frac{1}{2} \sin \left(x^{2}+1\right)+C} \\ {\text { (c) } \frac{1}{2} \sin \left(x^{2}+1\right)+C} & {\text { (d) }-2 x \sin \left(x^{2}+1\right)+C}\end{array} $$

In Exercises 1–6, identify and for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int \ln 5 x d x $$

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{2} e^{-x} d x $$

Using Two Methods In Exercises \(5-10\) , evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hopital's Rule. $$ \lim _{x \rightarrow 4} \frac{3(x-4)}{x^{2}-16} $$

Using Partial Fractions In Exercises \(5-22,\) use partial fractions to find the indefinite integral. $$ \int \frac{1}{x^{2}-9} d x $$

Use a table of integrals with forms involving the trigonometric functions to find the indefinite integral. \(\int \cos ^{4} 3 x d x\)

Using Trigonometric Substitution In Exercises \(5-8,\) find the indefinite integral using the substitution \(x=4 \sin \theta .\) $$ \int \frac{1}{\left(16-x^{2}\right)^{3 / 2}} d x $$

Finding an Indefinite Integral Involving Sine and cosine In Exercises \(1-12,\) find the indefinite integral. $$ \int \sin ^{3} x \cos ^{2} x d x $$

Choosing a Formula In Exercises \(5-14\) , select the basic integration formula you can use to find the integral, and identify \(u\) and \(a\) when appropriate. $$ \int(5 x-3)^{4} d x $$

In Exercises 1–6, identify and for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int x \sec ^{2} x d x $$

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{\infty} \cos x d x $$

Using Two Methods In Exercises \(5-10\) , evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hopital's Rule. $$ \lim _{x \rightarrow-4} \frac{2 x^{2}+13 x+20}{x+4} $$

Using Partial Fractions In Exercises \(5-22,\) use partial fractions to find the indefinite integral. $$ \int \frac{2}{9 x^{2}-1} d x $$

Use a table of integrals with forms involving the trigonometric functions to find the indefinite integral. \(\int \frac{\sin ^{4} \sqrt{x}}{\sqrt{x}} d x\)

Using Trigonometric Substitution In Exercises \(5-8,\) find the indefinite integral using the substitution \(x=4 \sin \theta .\) $$ \int \frac{4}{x^{2} \sqrt{16-x^{2}}} d x $$

Finding an Indefinite Integral Involving Sine and cosine In Exercises \(1-12,\) find the indefinite integral. $$ \int \cos ^{3} \frac{x}{3} d x $$

Choosing a Formula In Exercises \(5-14\) , select the basic integration formula you can use to find the integral, and identify \(u\) and \(a\) when appropriate. $$ \int \frac{2 t+1}{t^{2}+t-4} d t $$

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{-\infty}^{\infty} \frac{\sin x}{4+x^{2}} d x $$

Using Two Methods In Exercises \(5-10\) , evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hopital's Rule. $$ \lim _{x \rightarrow 6} \frac{\sqrt{x+10}-4}{x-6} $$

Using Partial Fractions In Exercises \(5-22,\) use partial fractions to find the indefinite integral. $$ \int \frac{5}{x^{2}+3 x-4} d x $$

Use a table of integrals with forms involving the trigonometric functions to find the indefinite integral. \(\int \frac{1}{\sqrt{x}(1-\cos \sqrt{x})} d x\)

Using Trigonometric Substitution In Exercises \(5-8,\) find the indefinite integral using the substitution \(x=4 \sin \theta .\) $$ \int \frac{\sqrt{16-x^{2}}}{x} d x $$