91Ó°ÊÓ

Evaluate the integrals in Exercises \(67-80\) $$ \int \frac{d y}{y^{2}+6 y+10} $$

L'Hopital's Rule does not help with the limits in Exercises \(67-74 .\) Try it - you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow \infty} \frac{e^{x^{2}}}{x e^{x}}$$

The region in the first quadrant bounded by the coordinate axes, the line \(y=3,\) and the curve \(x=2 / \sqrt{y+1}\) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid.

Evaluate the integrals in Exercises \(67-80\) $$ \int_{1}^{2} \frac{8 d x}{x^{2}-2 x+2} $$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{3}\left(\left(\frac{x+1}{x-1}\right)^{\ln 3}\right)$$

The region between the curve \(y=\sqrt{\cot x}\) and the \(x\) -axis from \(x=\pi / 6\) to \(x=\pi / 2\) is revolved about the \(x\) -axis to generate a solid. Find the volume of the solid.

Evaluate the integrals in Exercises \(67-74\) in terms of a. inverse hyperbolic functions. b. natural logarithms. $$\int_{1}^{e} \frac{d x}{x \sqrt{1+(\ln x)^{2}}}$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{5} \sqrt{\left(\frac{7 x}{3 x+2}\right)^{\ln 5}}$$

L'Hopital's Rule does not help with the limits in Exercises \(67-74 .\) Try it - you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow 0^{+}} \frac{x}{e^{-1 / x}}$$

Evaluate the integrals in Exercises \(67-80\) $$ \int_{2}^{4} \frac{2 d x}{x^{2}-6 x+10} $$