Problem 32
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. $$ \int_{3} \frac{1}{x^{2} \sqrt{x^{2}-9}} d x $$
Problem 32
Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{\ln 2 x}{x^{2}} d x $$
Problem 32
Use partial fractions to find the indefinite integral. $$ \int \frac{3 x}{x^{2}-6 x+9} d x $$
Problem 32
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 \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 1.5) $$
Problem 33
Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x \sqrt{x-1} d x $$
Problem 33
Consider the region satisfying the inequalities. Find the area of the region. $$ y \leq \frac{1}{x^{2}}, y \geq 0, x \geq 1 $$
Problem 33
$$ \text { Evaluate the definite integral. } $$ $$ \int_{4}^{5} \frac{1}{9-x^{2}} d x $$
Problem 34
$$ \text { Evaluate the definite integral. } $$ $$ \int_{0}^{1} \frac{3}{2 x^{2}+5 x+2} d x $$
Problem 34
Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{x}{\sqrt{x-1}} d x $$
Problem 34
Consider the region satisfying the inequalities. Find the area of the region. $$ y \leq e^{-x}, y \geq 0, x \geq 0 $$
