Chapter 8

In Exercises \(1-8,\) determine which are probability density functions and justify your answer. $$ f(x)=\frac{1}{18} x \text { over }[4,8] $$

The integrals converge. Evaluate the integrals without using tables. $$\int_{0}^{\infty} \frac{d x}{x^{2}+1}$$

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{d x}{x \sqrt{x-3}}\)

Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{5 x-13}{(x-3)(x-2)}$$

The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula (( true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula ( \(\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$ \int_{1}^{2} x d x $$

Evaluate the integrals. $$\int \frac{d x}{\sqrt{9+x^{2}}}$$

Evaluate the integrals in Exercises \(1-22\) $$ \int \cos 2 x d x $$

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int x \sin \frac{x}{2} d x $$

The integrals in Exercises \(1-40\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int_{0}^{1} \frac{16 x}{8 x^{2}+2} d x $$

The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula (( true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula ( \(\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$ \int_{1}^{3}(2 x-1) d x $$

In Exercises \(1-8,\) determine which are probability density functions and justify your answer. $$ f(x)=\frac{1}{2}(2-x) \text { over }[0,2] $$

The integrals converge. Evaluate the integrals without using tables. $$\int_{1}^{\infty} \frac{d x}{x^{1.001}}$$

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{d x}{x \sqrt{x+4}}\)

Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{5 x-7}{x^{2}-3 x+2}$$

Evaluate the integrals. $$\int \frac{3 d x}{\sqrt{1+9 x^{2}}}$$

Evaluate the integrals in Exercises \(1-22\) $$ \int_{0}^{\pi} 3 \sin \frac{x}{3} d x $$

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int \theta \cos \pi \theta d \theta $$

The integrals in Exercises \(1-40\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int \frac{x^{2}}{x^{2}+1} d x $$

The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula (( true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula ( \(\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$ \int_{-1}^{1}\left(x^{2}+1\right) d x $$

In Exercises \(1-8,\) determine which are probability density functions and justify your answer. $$ f(x)=2^{x} \text { over }\left[0, \frac{\ln (1+\ln 2)}{\ln 2}\right] $$

The integrals converge. Evaluate the integrals without using tables. $$\int_{0}^{1} \frac{d x}{\sqrt{x}}$$

Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{x+4}{(x+1)^{2}}$$

Evaluate the integrals. $$\int_{-2}^{2} \frac{d x}{4+x^{2}}$$

Evaluate the integrals in Exercises \(1-22\) $$ \int \cos ^{3} x \sin x d x $$

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int t^{2} \cos t d t $$

The integrals in Exercises \(1-40\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int(\sec x-\tan x)^{2} d x $$

The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula (( true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula ( \(\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$ \int_{-2}^{0}\left(x^{2}-1\right) d x $$

In Exercises \(1-8,\) determine which are probability density functions and justify your answer. $$ f(x)=x-1 \text { over }[0,1+\sqrt{3}] $$

The integrals converge. Evaluate the integrals without using tables. $$\int_{0}^{4} \frac{d x}{\sqrt{4-x}}$$

Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{2 x+2}{x^{2}-2 x+1}$$

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{x d x}{(2 x+3)^{3 / 2}}\)

Evaluate the integrals. $$\int_{0}^{2} \frac{d x}{8+2 x^{2}}$$

Evaluate the integrals in Exercises \(1-22\) $$ \int \sin ^{4} 2 x \cos 2 x d x $$

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int x^{2} \sin x d x $$

The integrals in Exercises \(1-40\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int_{\pi / 4}^{\pi / 3} \frac{d x}{\cos ^{2} x \tan x} $$

The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula (( true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula ( \(\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$ \int_{0}^{2}\left(t^{3}+t\right) d t $$

In Exercises \(1-8,\) determine which are probability density functions and justify your answer. $$ f(x)=\left\\{\begin{array}{ll}{\frac{1}{x^{2}}} & {x \geq 1} \\ {0} & {x<1}\end{array}\right. $$

Use the table of integrals at the back of the book to evaluate the integrals. \(\int x \sqrt{2 x-3} d x\)

Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{z+1}{z^{2}(z-1)}$$

Evaluate the integrals. $$\int_{0}^{3 / 2} \frac{d x}{\sqrt{9-x^{2}}}$$

Evaluate the integrals in Exercises \(1-22\) $$ \int \sin ^{3} x d x $$

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int_{1}^{2} x \ln x d x $$

The integrals in Exercises \(1-40\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int \frac{1-x}{\sqrt{1-x^{2}}} d x $$

The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula (( true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula ( \(\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$ \int_{-1}^{1}\left(t^{3}+1\right) d t $$

In Exercises \(1-8,\) determine which are probability density functions and justify your answer. $$ f(x)=\left\\{\begin{array}{ll}{\frac{8}{\pi\left(4+x^{2}\right)}} & {x \geq 0} \\\ {0} & {x<0}\end{array}\right. $$

Use the table of integrals at the back of the book to evaluate the integrals. \(\int x(7 x+5)^{3 / 2} d x\)

The integrals converge. Evaluate the integrals without using tables. $$\int_{-8}^{1} \frac{d x}{x^{1 / 3}}$$

Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{z}{z^{3}-z^{2}-6 z}$$

Evaluate the integrals. $$\int_{0}^{1 / 2 \sqrt{2}} \frac{2 d x}{\sqrt{1-4 x^{2}}}$$

Evaluate the integrals in Exercises \(1-22\) $$ \int \cos ^{3} 4 x d x $$