Chapter 7

Verify the integration formulas. \(\int \frac{\tan ^{-1} x}{x^{2}} d x=\ln x-\frac{1}{2} \ln \left(1+x^{2}\right)-\frac{\tan ^{-1} x}{x}+C\)

Evaluate the integrals in Exercises \(93-106.\) $$\int_{1}^{e} \frac{2 \ln 10 \log _{10} x}{x} d x$$

Verify the integration formulas. \(\int x^{3} \cos ^{-1} 5 x d x=\frac{x^{4}}{4} \cos ^{-1} 5 x+\frac{5}{4} \int \frac{x^{4} d x}{\sqrt{1-25 x^{2}}}\)

Evaluate the integrals in Exercises \(93-106.\) $$\int_{0}^{2} \frac{\log _{2}(x+2)}{x+2} d x$$

Verify the integration formulas. \(\int\left(\sin ^{-1} x\right)^{2} d x=x\left(\sin ^{-1} x\right)^{2}-2 x+2 \sqrt{1-x^{2}} \sin ^{-1} x+C\)

Verify the integration formulas. \(\int \ln \left(a^{2}+x^{2}\right) d x=x \ln \left(a^{2}+x^{2}\right)-2 x+2 a \tan ^{-1} \frac{x}{a}+C\)

Evaluate the integrals in Exercises \(93-106.\) $$\int_{0}^{9} \frac{2 \log _{10}(x+1)}{x+1} d x$$

Solve the initial value problems. $$\frac{d y}{d x}=\frac{1}{\sqrt{1-x^{2}}}, \quad y(0)=0$$

Evaluate the integrals in Exercises \(93-106.\) $$\int_{2}^{3} \frac{2 \log _{2}(x-1)}{x-1} d x$$

Solve the initial value problems. $$\frac{d y}{d x}=\frac{1}{x^{2}+1}-1, \quad y(0)=1$$

Evaluate the integrals in Exercises \(93-106.\) $$\int \frac{d x}{x \log _{10} x}$$

Solve the initial value problems. $$\frac{d y}{d x}=\frac{1}{x \sqrt{x^{2}-1}}, \quad x>1 ; \quad y(2)=\pi$$

Evaluate the integrals in Exercises \(93-106.\) $$\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$$

Solve the initial value problems. $$\frac{d y}{d x}=\frac{1}{1+x^{2}}-\frac{2}{\sqrt{1-x^{2}}}, \quad y(0)=2$$

Evaluate the integrals in Exercises \(107-110.\) $$\int_{1}^{\ln x} \frac{1}{t} d t, \quad x>1$$

You are sitting in a classroom next to the wall looking at the blackboard at the front of the room. The blackboard is \(4 \mathrm{m}\) long and starts \(1 \mathrm{m}\) from the wall you are sitting next to. a. Show that your viewing angle is \(\alpha=\cot ^{-1} \frac{x}{5}-\cot ^{-1} x\) if you are \(x\) m from the front wall. b. Find \(x\) so that \(\alpha\) is as large as possible.

The region between the curve \(y=\sec ^{-1} x\) and the \(x\) -axis from \(x=1\) to \(x=2\) (shown here) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid.

Evaluate the integrals in Exercises \(107-110.\) $$\frac{1}{\ln a} \int_{1}^{x} \frac{1}{t} d t, \quad x>0$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(x+1)^{x}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=x^{2}+x^{2 x}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\sqrt{t})^{t}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=t^{\sqrt{t}}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\sin x)^{x}$$

Use the identity \(\csc ^{-1} u=\frac{\pi}{2}-\sec ^{-1} u\) to derive the formula for the derivative of \(\csc ^{-1} u\) in Table 7.3 from the formula for the derivative of \(\sec ^{-1} u\).

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=x^{\sin x}$$

Derive the formula \(\frac{d y}{d x}=\frac{1}{1+x^{2}}\) for the derivative of \(y=\tan ^{-1} x\) by differentiating both sides of the equivalent equation tan \(y=x\).

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sin x^{x}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\ln x)^{\ln x}$$

Use the identity \(\cot ^{-1} u=\frac{\pi}{2}-\tan ^{-1} u\) to derive the formula for the derivative of \(\cot ^{-1} u\) in Table 7.3 from the formula for the derivative of \(\tan ^{-1} u\).

Find the absolute maximum and minimum values of \(f(x)=\) \(e^{x}-2 x\) on [0,1]

What is special about the functions \(f(x)=\sin ^{-1} \frac{x-1}{x+1}, \quad x \geq 0, \quad\) and \(\quad g(x)=2 \tan ^{-1} \sqrt{x} ?\) Explain.

Where does the periodic function \(f(x)=2 e^{\sin (x / 2)}\) take on its extreme values and what are these values?

What is special about the functions \(f(x)=\sin ^{-1} \frac{1}{\sqrt{x^{2}+1}} \quad\) and \(\quad g(x)=\tan ^{-1} \frac{1}{x} ?\) Explain.

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. Let \(f(x)=x e^{-x}\) a. Find all absolute extreme values for \(f\) b. Find all inflection points for \(f\)

Let \(f(x)=\frac{e^{x}}{1+e^{2 x}}\) a. Find all absolute extreme values for \(f\) b. Find all inflection points for \(f\)

Find the volumes of the solids. The solid lies between planes perpendicular to the \(x\) -axis at \(x=-1\) and \(x=1 .\) The cross-sections perpendicular to the \(x\) -axis are a. circles whose diameters stretch from the curve \(y=\) \(-1 / \sqrt{1+x^{2}}\) to the curve \(y=1 / \sqrt{1+x^{2}}\). b. vertical squares whose base edges run from the curve \(y=\) \(-1 / \sqrt{1+x^{2}}\) to the curve \(y=1 / \sqrt{1+x^{2}}\).

Graph \(f(x)=(x-3)^{2} e^{x}\) and its first derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime} .\) Identify significant points on the graphs with calculus, as necessary.

Find the volumes of the solids. The solid lies between planes perpendicular to the \(x\) -axis at \(x=-\sqrt{2} / 2\) and \(x=\sqrt{2} / 2 .\) The cross-sections perpendicular to the \(x\) -axis are a. circles whose diameters stretch from the \(x\) -axis to the curve \(y=2 / \sqrt[4]{1-x^{2}}\). b. squares whose diagonals stretch from the \(x\) -axis to the curve \(y=2 / \sqrt[4]{1-x^{2}}\).

Find the area of the "triangular" region in the first quadrant that is bounded above by the curve \(y=e^{2 x},\) below by the curve \(y=e^{x},\) and on the right by the line \(x=\ln 3.\)

Find the values of the following. a. \(\sec ^{-1} 1.5\) b. \(\csc ^{-1}(-1.5)\) c. \(\cot ^{-1} 2\)

Find the area of the "triangular" region in the first quadrant that is bounded above by the curve \(y=e^{x / 2},\) below by the curve \(y=e^{-x / 2},\) and on the right by the line \(x=2 \ln 2\)

Find the values of the following. a. \(\sec ^{-1}(-3)\) b. \(\csc ^{-1} 1.7\) c. \(\cot ^{-1}(-2)\)

Find the domain and range of each composite function. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see. a. \(y=\tan ^{-1}(\tan x)\) b. \(y=\tan \left(\tan ^{-1} x\right)\)

Find the area of the surface generated by revolving the curve \(x=\left(e^{y}+e^{-y}\right) / 2,0 \leq y \leq \ln 2,\) about the \(y\) -axis.

Find the domain and range of each composite function. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see. a. \(y=\sin ^{-1}(\sin x)\) b. \(y=\sin \left(\sin ^{-1} x\right)\)

Find the length of each curve. $$y=\frac{1}{2}\left(e^{x}+e^{-x}\right) \text { from } x=0 \text { to } x=1$$

Find the domain and range of each composite function. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see. a. \(y=\cos ^{-1}(\cos x)\) b. \(y=\cos \left(\cos ^{-1} x\right)\)

Find the length of each curve. $$y=\ln \left(e^{x}-1\right)-\ln \left(e^{x}+1\right) \text { from } x=\ln 2 \text { to } x=\ln 3$$

Use your graphing utility. Graph \(y=4 x /\left(x^{2}+1\right),\) known as Newton's serpentine. Then graph \(y=2 \sin \left(2 \tan ^{-1} x\right)\) in the same graphing window. What do you see? Explain.

Find the length of each curve. $$y=\ln (\csc x) \text { from } x=\pi / 6 \text { to } x=\pi / 4$$