Chapter 5

The table gives the values of a function obtained from an experiment. Use them to estimate \(\int_{3}^{9} f(x) d x\) using three equal subintervals with (a) right endpoints, (b) left end- points, and (c) midpoints. If the function is known to be an increasing function, can you say whether your estimates are less than or greater than the exact value of the integral? $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline f(x) & {-3.4} & {-2.1} & {-0.6} & {0.3} & {0.9} & {1.4} & {1.8} \\\ \hline\end{array}$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{2 \pi}^{\infty} \sin \theta d \theta\)

Evaluate the integral. $$\int \frac{a x}{x^{2}-b x} d x$$

Evaluate the integral. \(\int_{1}^{2}(1+2 y)^{2} d y\)

Evaluate the indefinite integral. \(\int(3 x-2)^{20} d x\)

Evaluate the integral. \(\int \ln \sqrt[3]{x} d x\)

9-12 Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. $$\int_{2}^{10} \sqrt{x^{3}+1} d x, \quad n=4$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{-\infty}^{\infty}\left(y^{3}-3 y^{2}\right) d y\)

Evaluate the integral. $$\int \frac{1}{(x+a)(x+b)} d x$$

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

Evaluate the indefinite integral. \(\int(3 t+2)^{2.4} d t\)

Evaluate the integral. \(\int p^{5} \ln p d p\)

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. $$\int_{0}^{\pi / 2} \cos ^{4} x d x, \quad n=4$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{-\infty}^{\infty} x e^{-x^{2}} d x\)

Use the Table of Integrals on Reference Pages \(6-10\) to evaluate the integral. \(\int \sin ^{2} x \cos x \ln (\sin x) d x\)

Evaluate the integral. $$\int_{0}^{1} \frac{2}{2 x^{2}+3 x+1} d x$$

Evaluate the integral. \(\int_{1}^{9} \frac{x-1}{\sqrt{x}} d x\)

Evaluate the indefinite integral. \(\int \sin \pi t d t\)

Evaluate the integral. \(\int e^{2 \theta} \sin 3 \theta d \theta\)

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. $$\int_{0}^{1} \sin \left(x^{2}\right) d x, \quad n=5$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{1}^{\infty} \frac{e^{-\sqrt{x}}}{\sqrt{x}} d x\)

Evaluate the integral. $$\int_{0}^{1} \frac{x^{3}-4 x-10}{x^{2}-x-6} d x$$

Evaluate the integral. \(\int_{-1}^{1} t(1-t)^{2} d t\)

Evaluate the indefinite integral. \(\int e^{x} \cos \left(e^{x}\right) d x\)

Evaluate the integral. \(\int e^{-\theta} \cos 2 \theta d \theta\)

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. $$\int_{1}^{5} x^{2} e^{-x} d x, \quad n=4$$

Evaluate the integral. $$\int_{1}^{2} \frac{4 y^{2}-7 y-12}{y(y+2)(y-3)} d y$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{1}^{\infty} \frac{x+1}{x^{2}+2 x} d x\)

Use the Table of Integrals on Reference Pages \(6-10\) to evaluate the integral. \(\int \frac{e^{x}}{3-e^{2 x}} d x\)

Evaluate the integral. \(\int_{0}^{1} x(\sqrt[3]{x}+\sqrt[4]{x}) d x\)

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

Evaluate the integral. \(\int_{0}^{\pi} t \sin 3 t d t\)

Drug pharmacokinetics During testing of a new drug, researchers measured the plasma drug concentration of each test subject at 10-minute intervals. The average concentrations \(C(t)\) are shown in the table, where t is measured in minutes and \(C\) is measured in \(\mu g / m L .\) Use the Midpoint Rule to estimate the integral \(\int_{0}^{100} C(t) d t\) State the units. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t & {0} & {10} & {20} & {30} & {40} & {50} \\ \hline C(t) & {0} & {1.3} & {1.8} & {2.2} & {2.4} & {2.5} \\\ \hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|c|}\hline t & {60} & {70} & {80} & {90} & {100} \\\ \hline C(t) & {2.4} & {2.3} & {2.0} & {1.6} & {1.1} \\ \hline\end{array}$$

Evaluate the integral. $$\int \frac{x^{2}+2 x-1}{x^{3}-x} d x$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{-\infty}^{\infty} \cos \pi t d t\)

Evaluate the integral. \(\int_{0}^{\pi / 4} \sec \theta \tan \theta d \theta\)

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

Evaluate the integral. \(\int_{0}^{1}\left(x^{2}+1\right) e^{-x} d x\)

Salicylic acid pharmacokinetics In the study cited in Example \(5,\) the metabolite salicylic acid (SA) was rapidly formed and peak SA levels of about 4.2\(\mu g / m L\) were reached after an hour. The concentration of SA was modeled by the function $$C(t)=11.4 t e^{-t}$$ where \(t\) is measured in hours and \(C\) is measured in \(\mu \mathrm{g} / \mathrm{mL}\) . Use the Midpoint Rule with eight subintervals to estimate the integral \(\int_{0}^{4} C(t) d t .\) State the units.

Make a substitution to express the integrand as a rational function and then evaluate the integral. $$\int_{9}^{16} \frac{\sqrt{x}}{x-4} d x$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{0}^{\infty} s e^{-5 s} d s\)

Evaluate the integral. \(\int_{0}^{\pi / 4} \sec ^{2} t d t\)

Evaluate the indefinite integral. \(\int \frac{d x}{5-3 x}\)

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

15-18 Express the limit as a definite integral on the given interval. $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} x_{i} \ln \left(1+x_{i}^{2}\right) \Delta x, \quad[2,6]$$

\(15-17\) Use Definition 2 to find an expression for the area under the graph of \(f\) as a limit. Do not evaluate the limit. $$f(x)=x^{2}+\sqrt{1+2 x}, 4 \leqq x \leq 7$$

Make a substitution to express the integrand as a rational function and then evaluate the integral. $$\int \frac{d x}{2 \sqrt{x+3}+x}$$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{-\infty}^{6} r e^{r / 3} d r\)

Evaluate the integral. \(\int_{1}^{18} \sqrt{\frac{3}{z}} d z\)

Evaluate the indefinite integral. \(\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x\)