91Ó°ÊÓ

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

Suppose that a cup of soup cooled from \(90^{\circ} \mathrm{C}\) to \(60^{\circ} \mathrm{C}\) after 10 min in a room whose temperature was \(20^{\circ} \mathrm{C} .\) Use Newton's Law of Cooling to answer the following questions. a. How much longer would it take the soup to cool to \(35^{\circ} \mathrm{C} ?\) b. Instead of being left to stand in the room, the cup of \(90^{\circ} \mathrm{C}\) soup is put in a freezer whose temperature is \(-15^{\circ} \mathrm{C}\). How long will it take the soup to cool from \(90^{\circ} \mathrm{C}\) to \(35^{\circ} \mathrm{C} ?\)

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

Evaluate the integrals. $$\int \sinh 2 x \, d x$$

Evaluate the integrals. $$\int_{1 / 10}^{10} \frac{\log _{10}(10 x)}{x} d x$$

An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at \(65^{\circ} \mathrm{F} .\) After 10 min, the beam warmed to \(35^{\circ} \mathrm{F}\) and after another \(10 \mathrm{min}\) it was \(50^{\circ} \mathrm{F} .\) Use Newton's Law of Cooling to estimate the beam's initial temperature.

Evaluate the integrals. $$\int \sinh \frac{x}{5} d x$$

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

Evaluate the integrals. $$\int 6 \cosh \left(\frac{x}{2}-\ln 3\right) d x$$

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