Chapter 12

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x^{4} \ln x d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{9} \frac{1}{\sqrt{9-x}} d x $$

Present Value In Exercises 25 and 26, use a program similar to the Simpson's Rule program on page 906 with \(n=8\) to approximate the present value of the income \(c(t)\) over \(t_{1}\) years at the given annual interest rate \(r\). Then use the integration capabilities of a graphing utility to approximate the present value. Compare the results. (Present value is defined in Section 12.1.) $$ c(t)=6000+200 \sqrt{t}, r=7 \%, t_{1}=4 $$

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

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{e^{1 / t}}{t^{2}} d t $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{2} \frac{x}{\sqrt{4-x^{2}}} d x $$

Present Value In Exercises 25 and 26, use a program similar to the Simpson's Rule program on page 906 with \(n=8\) to approximate the present value of the income \(c(t)\) over \(t_{1}\) years at the given annual interest rate \(r\). Then use the integration capabilities of a graphing utility to approximate the present value. Compare the results. (Present value is defined in Section 12.1.) $$ c(t)=200,000+15,000 \sqrt[3]{t}, r=10 \%, t_{1}=8 $$

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

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{1}{x(\ln x)^{3}} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} \frac{1}{x^{2}} d x $$

Marginal Analysis In Exercises 27 and 28, use a program similar to the Simpson's Rule program on page 906 with \(n=4\) to approximate the change in revenue from the marginal revenue function \(d R / d x .\) In each case, assume that the number of units sold \(x\) increases from 14 to 16 . $$ \frac{d R}{d x}=5 \sqrt{8000-x^{3}} $$

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

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x(\ln x)^{2} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} \frac{1}{x} d x $$

Marginal Analysis In Exercises 27 and 28, use a program similar to the Simpson's Rule program on page 906 with \(n=4\) to approximate the change in revenue from the marginal revenue function \(d R / d x .\) In each case, assume that the number of units sold \(x\) increases from 14 to 16 . $$ \frac{d R}{d x}=50 \sqrt{x} \sqrt{20-x} $$

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

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \ln 3 x d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x $$

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 1) $$

Use partial fractions to find the indefinite integral. $$ \int \frac{2 x-3}{(x-1)^{2}} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{(\ln x)^{2}}{x} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{2} \frac{1}{(x-1)^{4 / 3}} d x $$

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 2) $$

Use the table of integrals in this section to find the indefinite integral. $$ \int \frac{2 x}{(1-3 x)^{2}} d x $$

Use partial fractions to find the indefinite integral. $$ \int \frac{x^{4}}{(x-1)^{3}} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{1}{x \ln x} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{3}^{4} \frac{1}{\sqrt{x^{2}-9}} d x $$

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 4) $$

Use partial fractions to find the indefinite integral. $$ \int \frac{3 x^{2}+3 x+1}{x\left(x^{2}+2 x+1\right)} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{\ln x}{x^{2}} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{3} \frac{1}{x^{2} \sqrt{x^{2}-9}} d x $$

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 1.5) $$

Use partial fractions to find the indefinite integral. $$ \int \frac{3 x}{x^{2}-6 x+9} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{\ln 2 x}{x^{2}} d x $$

Consider the region satisfying the inequalities. Find the area of the region. $$ y \leq \frac{1}{x^{2}}, y \geq 0, x \geq 1 $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{4}^{5} \frac{1}{9-x^{2}} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x \sqrt{x-1} d x $$

Consider the region satisfying the inequalities. Find the area of the region. $$ y \leq e^{-x}, y \geq 0, x \geq 0 $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{0}^{1} \frac{3}{2 x^{2}+5 x+2} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{x}{\sqrt{x-1}} d x $$

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=1, n=1 $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{2} x^{3} d x $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{1}^{5} \frac{x-1}{x^{2}(x+1)} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x(x+1)^{2} d x $$

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=2, n=4 $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} \frac{1}{x+1} d x $$

$$ \text { Evaluate the definite integral. } $$ $$ \int_{0}^{1} \frac{x^{2}-x}{x^{2}+x+1} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{x}{\sqrt{2+3 x}} d x $$

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=\frac{1}{2}, n=2 $$

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} e^{x^{3}} d x $$