Chapter 7

Solve the following problems. $$\frac{d y}{d x}=3 \cos 2 x+2 \sin 3 x, y(\pi / 2)=8$$

Evaluate the following integrals or state that they diverge. $$\int_{-\infty}^{\infty} \frac{d x}{x^{2}+2 x+5}$$

Find the Midpoint and Trapezoid Rule approximations to \(\int_{0}^{1} e^{-x} d x\) using \(n=50\) subintervals. Compute the relative error of each approximation.

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d v}{v\left(v^{2}+8\right)}$$

Evaluate the following integrals. $$\int \frac{d x}{\left(1+x^{2}\right)^{3 / 2}}$$

Evaluate the following integrals. $$\int \frac{y+1}{y^{3}+3 y^{2}-18 y} d y$$

Evaluate the following integrals. $$\int x \sec ^{-1} x d x, x \geq 1$$

Evaluate the following integrals. $$\int_{0}^{2} \frac{x(3 x+2)}{\sqrt{x^{3}+x^{2}+4}} d x$$

Find the general solution of the following equations. $$y^{\prime}(t)=3 y-4$$

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\infty} \frac{e^{u}}{e^{2 u}+1} d u$$

Evaluate the following integrals. $$\int \frac{d x}{x^{2} \sqrt{x^{2}+9}}$$

Evaluate the following integrals. $$\int \frac{6 x^{2}}{x^{4}-5 x^{2}+4} d x$$

Evaluate the following integrals. $$\int x \sin x \cos x d x$$

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

Find the general solution of the following equations. $$\frac{d y}{d x}=-y+2$$

Evaluate the following integrals or state that they diverge. $$\int_{-\infty}^{a} \sqrt{e^{x}} d x, a \text { real }$$

Evaluate the following integrals. $$\int \frac{d t}{t^{2} \sqrt{9-t^{2}}}$$

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

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

Evaluate the following integrals. $$\int \frac{d y}{y^{-1}+y^{-3}}$$

Find the general solution of the following equations. $$y^{\prime}(x)=-2 y-4$$

Evaluate the following integrals or state that they diverge. $$\int_{1}^{\infty} \frac{d v}{v(v+1)}$$

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. $$\int \sqrt{x^{2}+10 x} d x, x>0$$

Evaluate the following integrals. $$\int \frac{d x}{\sqrt{36-x^{2}}}$$

Evaluate the following integrals. $$\int \frac{x^{2}+12 x-4}{x^{3}-4 x} d x$$

Integrals of \(\sin x\) and \(\cos x\) Evaluate the following integrals. $$\int \sin ^{2} x \cos ^{4} x d x$$

Evaluate the following integrals. $$\int t^{2} e^{-t} d t$$

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

Find the general solution of the following equations. $$\frac{d y}{d t}=2 y+6$$

Evaluate the following integrals or state that they diverge. $$\int_{1}^{\infty} \frac{d x}{x^{2}(x+1)}$$

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. $$\int \sqrt{x^{2}-8 x} d x, x>8$$

Evaluate the following integrals. $$\int \frac{d x}{\sqrt{16+4 x^{2}}}$$

Evaluate the following integrals. $$\int \frac{z^{2}+20 z-15}{z^{3}+4 z^{2}-5 z} d z$$

Evaluate the following integrals. $$\int e^{3 x} \cos 2 x d x$$

Evaluate the following integrals. $$\int_{4}^{9} \frac{x^{5 / 2}-x^{1 / 2}}{x^{3 / 2}} d x$$

Solve the following problems. $$y^{\prime}(t)=3 y-6, y(0)=9$$

Evaluate the following integrals or state that they diverge. $$\int_{1}^{\infty} \frac{3 x^{2}+1}{x^{3}+x} d x$$

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. $$\int \frac{d x}{x^{2}+2 x+10}$$

Evaluate the following integrals. $$\int \frac{d x}{\sqrt{x^{2}-81}}, x>9$$

Evaluate the following integrals. $$\int e^{-x} \sin 4 x d x$$

Evaluate the following integrals. $$\int \frac{\sin t+\tan t}{\cos ^{2} t} d t$$

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. \(\int_{0}^{8} e^{-2 x} d x=\frac{1-e^{-16}}{2}\)

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

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

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

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

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 or state that they diverge. $$\int_{2}^{\infty} \frac{d x}{(x+2)^{2}}$$

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

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. $$\int \frac{d x}{x\left(x^{10}+1\right)}$$