91Ó°ÊÓ

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

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

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

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. $$y=x^{2}+x^{2 x}$$