Chapter 6

For each definite integral: a. Approximate it "by hand," using trapezoidal approximation with \(n=4\) trapezoids. Round calculations to three decimal places. b. Evaluate the integral exactly using antiderivatives, rounding to three decimal places. c. Find the actual error (the difference between the actual value and the approximation). d. Find the relative error (the actual error divided by the actual value, expressed as a percent). \(\int_{1}^{3} x^{2} d x\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{x \rightarrow \infty} \frac{1}{x^{2}} $$

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. \(\int e^{2 x} d x\)

For each definite integral: a. Approximate it "by hand," using trapezoidal approximation with \(n=4\) trapezoids. Round calculations to three decimal places. b. Evaluate the integral exactly using antiderivatives, rounding to three decimal places. c. Find the actual error (the difference between the actual value and the approximation). d. Find the relative error (the actual error divided by the actual value, expressed as a percent). \(\int_{1}^{2} x^{3} d x\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{b \rightarrow \infty}\left(\frac{1}{\sqrt{b}}-8\right) $$

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. \(\int x^{5} d x\)

For each definite integral: a. Approximate it "by hand," using trapezoidal approximation with \(n=4\) trapezoids. Round calculations to three decimal places. b. Evaluate the integral exactly using antiderivatives, rounding to three decimal places. c. Find the actual error (the difference between the actual value and the approximation). d. Find the relative error (the actual error divided by the actual value, expressed as a percent). \(\int_{2}^{4} \frac{1}{x} d x\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{b \rightarrow \infty}\left(1-2 e^{-5 b}\right) $$

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. \(\int(x+2) d x\)

For each definite integral: a. Approximate it "by hand," using trapezoidal approximation with \(n=4\) trapezoids. Round calculations to three decimal places. b. Evaluate the integral exactly using antiderivatives, rounding to three decimal places. c. Find the actual error (the difference between the actual value and the approximation). d. Find the relative error (the actual error divided by the actual value, expressed as a percent). \(\int_{1}^{3} \frac{1}{x} d x\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{b \rightarrow \infty}\left(3 e^{3 b}-4\right) $$

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. \(\int(x-1) d x\)

Approximate each integral using trapezoidal approximation "by hand" with the given value of \(n\). Round all calculations to three decimal places. \(\int_{0}^{1} \sqrt{1+x^{2}} d x, \quad n=3\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{x \rightarrow \infty}\left(2-e^{x / 2}\right) $$

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. \(\int \sqrt{x} d x\)

Approximate each integral using trapezoidal approximation "by hand" with the given value of \(n\). Round all calculations to three decimal places. \(\int_{0}^{1} \sqrt{1+x^{3}} d x, \quad n=3\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{x \rightarrow \infty}\left(1-e^{-x / 3}\right) $$

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. \(\int e^{-0.5 t} d t\)

Approximate each integral using trapezoidal approximation "by hand" with the given value of \(n\). Round all calculations to three decimal places. \(\int_{0}^{1} e^{-x^{2}} d x, \quad n=4\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{b \rightarrow \infty}(3+\ln b) $$

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. \(\int(x+3)^{4} d x\)

\(7-42 .\) Find each integral by using the integral table on the inside back cover. $$ \begin{aligned} &\int \frac{1}{9-x^{2}} d x\\\ &\text { [Hint: Use formula } 16 \text { with } a=3 . \end{aligned} $$

Approximate each integral using trapezoidal approximation "by hand" with the given value of \(n\). Round all calculations to three decimal places. \(\int_{0}^{1} e^{x^{2}} d x, \quad n=4\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{b \rightarrow \infty}\left(2-\ln b^{2}\right) $$

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. \(\int(x-5)^{6} d x\)

Find each integral by using the integral table on the inside back cover. $$ \begin{aligned} &\int \frac{1}{x^{2}-25} d x\\\ &\text { [Hint: Use formula } 15 \text { with } a=5 . \end{aligned} $$

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{x \rightarrow-\infty} \frac{1}{x^{3}} $$

Use integration by parts to find each integral. \(x e^{2 x} d x\)

Use a graphing calculator trapezoidal approximation program from the Internet (see page 418 ) to approximate each integral. Use the following values for the numbers of intervals: \(10,50,100,200,500 .\) Then give an estimate for the value of the definite integral, keeping as many decimal places as the last two approximations agree (when rounded). $$ \int_{0}^{1} \ln \left(x^{2}+1\right) d x $$

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{x \rightarrow-\infty} \frac{1}{x^{2}} $$

Use integration by parts to find each integral. \(\int x e^{3 x} d x\)

Use a graphing calculator trapezoidal approximation program from the Internet (see page 418 ) to approximate each integral. Use the following values for the numbers of intervals: \(10,50,100,200,500 .\) Then give an estimate for the value of the definite integral, keeping as many decimal places as the last two approximations agree (when rounded). $$ \int_{-1}^{1} \sqrt{16+9 x^{2}} d x $$

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{a \rightarrow-\infty} e^{2 a} $$

Use integration by parts to find each integral. \(\int x^{5} \ln x d x\)

Use a graphing calculator trapezoidal approximation program from the Internet (see page 418 ) to approximate each integral. Use the following values for the numbers of intervals: \(10,50,100,200,500 .\) Then give an estimate for the value of the definite integral, keeping as many decimal places as the last two approximations agree (when rounded). $$ \int_{-1}^{1} \sqrt{25-9 x^{2}} d x $$

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{a \rightarrow-\infty} e^{-3 a} $$

Use integration by parts to find each integral. \(\int x^{4} \ln x d x\)

Find each integral by using the integral table on the inside back cover. $$ \int \frac{x}{\sqrt{1-x}} d x \text { [Hint: Use formula 13. } $$

Use a graphing calculator trapezoidal approximation program from the Internet (see page 418 ) to approximate each integral. Use the following values for the numbers of intervals: \(10,50,100,200,500 .\) Then give an estimate for the value of the definite integral, keeping as many decimal places as the last two approximations agree (when rounded). $$ \int_{-1}^{1} e^{x^{2}} d x $$

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{a \rightarrow-\infty} e^{-2 a} $$

Use integration by parts to find each integral. \(\int(x+2) e^{x} d x\)

Find each integral by using the integral table on the inside back cover. $$ \int \frac{1}{(2 x+1)(x+1)} d x $$

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{a \rightarrow-\infty} e^{\frac{1}{2} a} $$

Use integration by parts to find each integral. \(\int(x-1) e^{x} d x \quad\)

Find each integral by using the integral table on the inside back cover. $$ \int \frac{x}{(x+1)(x+2)} d x $$

Use integration by parts to find each integral. \(\int \sqrt{x} \ln x d x\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{x \rightarrow-\infty} \frac{1}{\left(x^{2}+1\right)^{2}} $$

Use integration by parts to find each integral. \(\int \sqrt[3]{x} \ln x d x\)

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{x \rightarrow-\infty} \frac{1}{e^{x}+e^{-x}} $$

Find each integral by using the integral table on the inside back cover. $$ \int \frac{1}{\sqrt{x^{2}-1}} d x $$