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 30
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. $$ \int_{0}^{2} \frac{1}{(x-1)^{4 / 3}} d x $$
Problem 30
Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{1}{x \ln x} d x $$
Problem 30
Use partial fractions to find the indefinite integral. $$ \int \frac{x^{4}}{(x-1)^{3}} d x $$
Problem 30
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 \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 2) $$
Problem 30
Use the table of integrals in this section to find the indefinite integral. $$ \int \frac{2 x}{(1-3 x)^{2}} d x $$
Problem 31
Use partial fractions to find the indefinite integral. $$ \int \frac{3 x^{2}+3 x+1}{x\left(x^{2}+2 x+1\right)} d x $$
Problem 31
Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{\ln x}{x^{2}} d x $$
Problem 31
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 \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 4) $$
Problem 31
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. $$ \int_{3}^{4} \frac{1}{\sqrt{x^{2}-9}} d x $$
