91Ó°ÊÓ

Differential Equation In Exercises 121 and 122, solve the differential equation. $$ \frac{d y}{d x}=x e^{a x^{2}} $$

Differential Equation In Exercises 121 and 122, solve the differential equation. $$ \frac{d y}{d x}=\left(e^{x}-e^{-x}\right)^{2} $$

Which is greater $$(\sqrt{n})^{\sqrt{n+1}}\( or \)(\sqrt{n+1}) \sqrt[1]{n}$$

Show that if \(x\) is positive, then $$\log _{e}\left(1+\frac{1}{x}\right)>\frac{1}{1+x}$$

Differential Equation In Exercises 123 and 124 , find the particular solution that satisfies the initial conditions. $$ \begin{array}{l}{f^{\prime \prime}(x)=\frac{1}{2}\left(e^{x}+e^{-x}\right)} \\\ {f(0)=1, f^{\prime}(0)=0}\end{array} $$

Differential Equation In Exercises 123 and 124 , find the particular solution that satisfies the initial conditions. $$ \begin{array}{l}{f^{\prime \prime}(x)=\sin x+e^{2 x}} \\ {f(0)=\frac{1}{4}, f^{\prime}(0)=\frac{1}{2}}\end{array} $$

Area In Exercises \(125-128\) , find the area of the region bounded by the graphs of the equations. Use a graphing utility to graph the region and verify your result. $$ y=e^{x}, y=0, x=0, x=5 $$

Area In Exercises \(125-128\) , find the area of the region bounded by the graphs of the equations. Use a graphing utility to graph the region and verify your result. $$ y=e^{-2 x}, y=0, x=-1, x=3 $$

Area In Exercises \(125-128\) , find the area of the region bounded by the graphs of the equations. Use a graphing utility to graph the region and verify your result. $$ y=x e^{-x^{2} / 4}, y=0, x=0, x=\sqrt{6} $$

Area In Exercises \(125-128\) , find the area of the region bounded by the graphs of the equations. Use a graphing utility to graph the region and verify your result. $$ y=e^{-2 x}+2, y=0, x=0, x=2 $$