Problem 27
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. $$ \int_{0}^{1} \frac{1}{x^{2}} d x $$
Problem 27
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 . $$ \frac{d R}{d x}=5 \sqrt{8000-x^{3}} $$
Problem 27
Use partial fractions to find the indefinite integral. $$ \int \frac{x+2}{x^{2}-4 x} d x $$
Problem 28
Use partial fractions to find the indefinite integral. $$ \int \frac{4 x^{2}+2 x-1}{x^{3}+x^{2}} d x $$
Problem 28
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. $$ \int_{0}^{1} \frac{1}{x} d x $$
Problem 28
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 . $$ \frac{d R}{d x}=50 \sqrt{x} \sqrt{20-x} $$
Problem 28
Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \ln 3 x d x $$
Problem 29
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. $$ \int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x $$
Problem 29
Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{(\ln x)^{2}}{x} d x $$
Problem 29
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 \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 1) $$
