91Ó°ÊÓ

Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. $$\int \frac{d x}{x^{2}+2 x+10}$$

Evaluate the following integrals. $$\int_{0}^{5} \frac{2}{x^{2}-4 x-32} d x$$

Solve the following problems. $$\frac{d y}{d x}=-y+2, y(0)=-2$$

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for \(n=4,8,16,\) and \(32 .\) The exact values of the integrals are given for computing the error. \(\int_{0}^{8} e^{-2 x} d x=\frac{1-e^{-16}}{2}\)

Evaluate the following integrals. $$\int \frac{d x}{\sqrt{1-2 x^{2}}}$$

Evaluate the following integrals. $$\int \frac{4+e^{-2 x}}{e^{3 x}} d x$$

Solve the following problems. $$y^{\prime}(t)=-2 y-4, y(0)=0$$

Hourly temperature data for Boulder, Colorado, San Francisco, California, Nantucket, Massachusetts, and Duluth, Minnesota, over a 12 hr period on the same day of January are shown in the figure. Assume that these data are taken from a continuous temperature function \(T(t) .\) The average temperature over the 12 -hr period is \(\bar{T}=\frac{1}{12} \int_{0}^{12} T(t) d t\). Find an accurate approximation to the average temperature over the 12 -hr period for Boulder. State your method.

Evaluate the following integrals. $$\int \frac{2-3 x}{\sqrt{1-x^{2}}} d x$$

Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. $$\int \frac{d x}{x\left(x^{10}+1\right)}$$