적분으로 정의한 로그·지수함수

Find the area of the region enclosed by $x=1$ and $y=5$ above $y=\ln \,x.$

For the following exercises, use the function $\ln \,x.$ If you are unable to find intersection points analytically, use a calculator.

[T] Find the arc length of $\ln \,x$ from $x=1$ to $x=2.$

Find the area between $\ln \,x$ and the x-axis from $x=1\,\text{to}\,x=2.$

Find the volume of the shape created when rotating this curve from $x=1\,\text{to}\,x=2$ around the x-axis, as pictured here.

[T] Find the surface area of the shape created when rotating the curve in the previous exercise from $x=1$ to $x=2$ around the x-axis.

Find the area of the hyperbolic quarter-circle enclosed by $x=2\,\text{and}\,y=2$ above $y=1/x.$

If you are unable to find intersection points analytically in the following exercises, use a calculator.

[T] Find the arc length of $y=1/x$ from $x=1\,\text{to}\,x=4.$

Find the area under $y=1/x$ and above the x-axis from $x=1\,\text{to}\,x=4.$

For the following exercises, verify the derivatives and antiderivatives.

$\frac{d}{dx}\ln \left(x+\sqrt{{x}^{2}+1}\right)=\frac{1}{\sqrt{1+{x}^{2}}}$

For the following exercises, verify the derivatives and antiderivatives.

$\frac{d}{dx}\ln \left(\frac{x-a}{x+a}\right)=\frac{2a}{({x}^{2}-{a}^{2})}$

For the following exercises, verify the derivatives and antiderivatives.

$\frac{d}{dx}\ln \left(\frac{1+\sqrt{1-{x}^{2}}}{x}\right)=-\frac{1}{x\sqrt{1-{x}^{2}}}$

For the following exercises, verify the derivatives and antiderivatives.

$\frac{d}{dx}\ln \left(x+\sqrt{{x}^{2}-{a}^{2}}\right)=\frac{1}{\sqrt{{x}^{2}-{a}^{2}}}$

For the following exercises, verify the derivatives and antiderivatives.

$\int \frac{dx}{x\,\ln (x)\ln (\ln \,x)}=\ln (\ln (\ln \,x))+C$