Chapter 5

Drug pharmacokinetics The plasma drug concentration of a new drug was modeled by the function \(C(t)=23 t e^{-2 t}\) where \(t\) is measured in hours and \(C\) in \(\mu \mathrm{g} / \mathrm{mL}\) . (a) What is the maximum drug concentration and when did it occur? (b) Calculate \(\int_{0}^{\infty} C(t) d t\) and explain its significance.

Evaluate the integral. \(\int_{0}^{\pi / 4} \frac{1+\cos ^{2} \theta}{\cos ^{2} \theta} d \theta\)

Evaluate the indefinite integral. \(\int \sqrt{\cot x} \csc ^{2} x d x\)

25-26 Express the integral as a limit of Riemann sums. Do not evaluate the limit. $$\int_{2}^{6} \frac{x}{1+x^{5}} d x$$

First make a substitution and then use integration by parts to evaluate the integral. \(\int x \ln (1+x) d x\)

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \sin ^{4} x d x\)

Spread of drug use In a study of the spread of illicit drug use from an enthusiastic user to a population of \(N\) users, the authors model the number of expected new users by the equation $$\gamma=\int_{0}^{\infty} \frac{c N\left(1-e^{-k_{l}}\right)}{k} e^{-\lambda t} d t$$ where \(c, k,\) and \(\lambda\) are positive constants. Evaluate this integral to express \(\gamma\) in terms of \(c, N, k,\) and \(\lambda\)

Evaluate the integral. \(\int_{1}^{2} \frac{(x-1)^{3}}{x^{2}} d x\)

Evaluate the indefinite integral. \(\int \frac{\cos (\pi / x)}{x^{2}} d x\)

Express the integral as a limit of Riemann sums. Do not evaluate the limit. $$\int_{1}^{10}(x-4 \ln x) d x$$

First make a substitution and then use integration by parts to evaluate the integral. \(\int \sin (\ln x) d x\)

Photosynthesis Much of the earth's photosynthesis occurs in the oceans. The rate of primary production depends on light intensity, measured as the flux of photons (that is, number of photons per unit area per unit time). Formonochromatic light, intensity decreases with water depth according to Beer's Law, which states that \(I(x)=e^{-k x},\) where \(x\) is water depth. A simple model for the relationship between rate of photosynthesis and light intensity is \(P(I)=a I,\) where \(a\) is a constant and \(P\) is measured as a mass of carbon fixed per volume of water, per unit time. Calculate \(\int_{0}^{\infty} P(I(x)) d x\) and interpret it.

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \frac{1}{\sqrt{1+\sqrt[3]{x}}} d x\)

Evaluate the integral. \(\int_{0}^{1 / \sqrt{3}} \frac{t^{2}-1}{t^{4}-1} d t\)

Evaluate the indefinite integral. \(\int e^{2 r} \sin \left(e^{2 r}\right) d r\)

(a) If \(n \geqslant 2\) is an integer, show that $$\int \sin ^{n} x d x=-\frac{1}{n} \cos x \sin ^{n-1} x+\frac{n-1}{n} \int \sin ^{n-2} x d x$$ This is called a reduction formula because the exponent \(n\) has been reduced to \(n-1\) and \(n-2 .\) (b) Use the reduction formula in part (a) to show that $$\int \sin ^{2} x d x=\frac{x}{2}-\frac{\sin 2 x}{4}+C$$ (c) Use parts (a) and (b) to evaluate \(\int \sin ^{4} x d x\) .

Dialysis treatment removes urea and other waste products from a patient's blood by diverting some of the bloodflow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in mg/min) is often well described by the equation $$c(t)=\frac{K}{V} c_{0} e^{-K t / V}$$ where \(K\) is the rate of flow of blood through the dialyzer (in \(\mathrm{mL} / \mathrm{min}\) , \(V\) is the volume of the patient's blood (in mL) and \(c_{0}\) is the amount of urea in the blood (in mg) at time \(t=0 .\) Evaluate the integral \(\int_{0}^{\infty} c(t) d t\) and interpret it.

Computer algebra systems sometimes need a helping hand from human beings. Try to evaluate \(\int(1+\ln x) \sqrt{1+(x \ln x)^{2}} d x\) with a computer algebra system. If it doesn't return an answer, make a substitution that changes the integral into one that the CAS can evaluate.

Evaluate the integral. \(\int_{0}^{2}|2 x-1| d x\)

Evaluate the indefinite integral. \(\int \frac{d t}{\cos ^{2} t \sqrt{1+\tan t}}\)

(a) Prove the reduction formula $$\int \cos ^{n} x d x=\frac{1}{n} \cos ^{n-1} x \sin x+\frac{n-1}{n} \int \cos ^{n-2} x d x$$ (b) Use part (a) to evaluate \(\int \cos ^{2} x d x\) (c) Use parts (a) and (b) to evaluate \(\int \cos ^{4} x d x\)

A manufacturer of lightbulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let \(F(t)\) be the fraction of the company's bulbs that burn out before \(t\) hours, so \(F(t)\) always lies between 0 and \(1 .\) (a) Make a rough sketch of what you think the graph of \(F\) might look like. (b) What is the meaning of the derivative \(r(t)=F^{\prime}(t) ?\) (c) What is the value of \(\int_{0}^{\infty} r(t) d t ?\) Why?

What is wrong with the equation? \(\int_{-1}^{3} \frac{1}{x^{2}} d x=\frac{x^{-1}}{-1} ]_{-1}^{3}=-\frac{4}{3}\)

Evaluate the indefinite integral. \(\int \sec ^{3} x \tan x d x\)

29-34 Evaluate the integral by interpreting it in terms of areas. $$\int_{0}^{3}\left(\frac{1}{2} x-1\right) d x$$

Use integration by parts to prove the reduction formula. \(\int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x\)

If \(\int_{-\infty}^{\infty} f(x) d x\) is convergent and \(a\) and \(b\) are real numbers, show that $$\int_{-\infty}^{a} f(x) d x+\int_{a}^{\infty} f(x) d x=\int_{-\infty}^{b} f(x) d x+\int_{b}^{\infty} f(x) d x$$

Evaluate the indefinite integral. \(\int x^{2} \sqrt{2+x} d x\)

Evaluate the integral by interpreting it in terms of areas. $$\int_{-2}^{2} \sqrt{4-x^{2}} d x$$

Use integration by parts to prove the reduction formula. \(\int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x\)

\(31-32\) Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \(y=\sin x, 0 \leqslant x \leqslant \pi\)

Evaluate the indefinite integral. \(\int x(2 x+5)^{8} d x\)

Evaluate the integral by interpreting it in terms of areas. $$\int_{-3}^{0}\left(1+\sqrt{9-x^{2}}\right) d x$$

For what values of \(p\) is the integral $$\int_{1}^{\infty} \frac{1}{x^{p}} d x$$ convergent? Evaluate the integral for those values of \(p\)

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \(y=\sec ^{2} x, 0 \leqslant x \leqslant \pi / 3\)

Evaluate the indefinite integral. \(\int \frac{e^{x}}{e^{x}+1} d x\)

Evaluate the integral by interpreting it in terms of areas. $$\int_{-1}^{3}(3-2 x) d x$$

Evaluate the integral, given that \(\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}\) \(\int_{0}^{\infty} e^{-x^{2} / 2} d x\)

Evaluate the indefinite integral. \(\int \frac{\sin 2 x}{1+\cos ^{2} x} d x\)

Evaluate the integral by interpreting it in terms of areas. $$\int_{-1}^{2}|x| d x$$

Salicylic acid pharmacokinetics In the article cited in Example 5 the authors also studied the formation and concentration of salicylic acid in the bloodstream of 10 volunteers. A model for the concentration is $$C(t)=11.4 t e^{-t}$$ where \(t\) is measured in hours and \(C\) in \(\mu \mathrm{g} / \mathrm{mL} .\) Calculate \(\int_{0}^{4} C(t) d t\) and include the units in your answer.

Evaluate the integral, given that \(\int_{0}^{\infty} e^{-x^{2}} d x=\frac{1}{2} \sqrt{\pi}\) \(\int_{0}^{\infty} x^{2} e^{-x^{2}} d x\)

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch. \(\int_{-\pi / 2}^{2 \pi} \cos x d x\)

Rumen microbial ecosystem The rumen is the first chamber in the stomach of ruminants such as cattle, sheep,and deer. Fermentation reactions by symbiotic organisms begin digesting plant matter in the rumen. If \(\mu\) is the fraction of matter entering or leaving the rumen per unit time in a model for continuous fermentation, the integral $$\int_{0}^{1} \mu e^{-\mu t}(1-t) d t$$ is the fraction of soluble material passing from the rumen in the first hour without being fermented. Evaluate this integral.

Evaluate the indefinite integral. \(\int \frac{\sin x}{1+\cos ^{2} x} d x\)

Evaluate the integral by interpreting it in terms of areas. $$\int_{0}^{10}|x-5| d x$$

\(35-36\) Verify by differentiation that the formula is correct. \(\int \cos ^{3} x d x=\sin x-\frac{1}{3} \sin ^{3} x+C\)

Gene regulation In Section 10.3 a model of gene regulation is analyzed and it is shown that the concentration of protein in a cell as a function of time is given by the equation $$p(t)=\frac{1}{2}-\frac{1}{2} e^{-t}(\sin t+\cos t)$$ The bioavailability of this protein is defined as the integral of this concentration over time. What is the bioavailability of the protein over the first unit of time?

Evaluate $$\int_{\pi}^{\pi} \sin ^{2} x \cos ^{4} x d x$$

Evaluate the indefinite integral. \(\int \frac{1+x}{1+x^{2}} d x\)