91Ó°ÊÓ

The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate. When necessary, use a substitution to reduce it to a standard form. $$ \int \frac{d \theta}{\cos \theta-1} $$

Use numerical integration to estimate the value of $$\pi=4 \int_{0}^{1} \frac{1}{1+x^{2}} d x$$

In Exercises \(35-68\) , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{0}^{\pi / 2} \tan \theta d \theta$$

Drug assimilation An average adult under age 60 years assimilates a 12 -hr cold medicine into his or her system at a rate modeled by $$\frac{d y}{d t}=6-\ln \left(2 t^{2}-3 t+3\right)$$ where \(y\) is measured in milligrams and \(t\) is the time in hours since the medication was taken. What amount of medicine is absorbed into a person's system over a 12 -hr period?

Evaluate the integrals in Exercises \(39-54\) $$ \int \frac{e^{t} d t}{e^{2 t}+3 e^{t}+2} $$

The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate. When necessary, use a substitution to reduce it to a standard form. $$ \begin{array}{l}{\int \frac{d x}{1+e^{x}}} \\ {\text {Hint} : \text { Use long division. }}\end{array} $$

In Exercises \(35-48\) , use an appropriate substitution and then a trigonometric substitution to evaluate the integrals. $$ \int \frac{d x}{x \sqrt{x^{2}-1}} $$

Evaluate the integrals. Some integrals do not require integration by parts. $$ \int x^{3} \sqrt{x^{2}+1} d x $$

Cholesterol levels The serum cholesterol levels of children aged 12 to 14 years follows a normal distribution with mean \(\mu=162\) mg/dl and standard deviation \(\sigma=28\) mg/dl. In a population of 1000 of these children, how many would you expect to have serum cholesterol levels between 165 and 193\(?\) between 148 and 167\(?\)

Evaluate the integrals. \(\int_{-\pi / 3}^{0} 2 \sec ^{3} x d x\)