91Ó°ÊÓ

Use the function \(\ln x\). If you are unable to find intersection points analytically, use a calculator. Find the area of the region enclosed by \(x=1\) and \(y=5\) above \(y=\ln x\)

Use the function \(\ln x\). If you are unable to find intersection points analytically, use a calculator. Find the arc length of \(\ln x\) from \(x=1\) to \(x=2\).

Use the function \(\ln x\). If you are unable to find intersection points analytically, use a calculator. Find the area between \(\ln x\) and the \(x\) -axis from \(x=1\) to \(x=2\)

Use the function \(\ln x\). If you are unable to find intersection points analytically, use a calculator. Find the area of the hyperbolic quarter-circle enclosed by \(x=2\) and \(y=2\) above \(y=1 / x\).

Use the function \(\ln x\). If you are unable to find intersection points analytically, use a calculator. Find the arc length of \(y=1 / x\) from \(x=1\) to \(x=4\)

Use the function \(\ln x\). If you are unable to find intersection points analytically, use a calculator. Find the area under \(y=1 / x\) and above the \(x\) -axis from \(x=1\) to \(x=4\)

Verify the derivatives and antiderivatives. $$ \frac{d}{d x} \ln \left(x+\sqrt{x^{2}+1}\right)=\frac{1}{\sqrt{1+x^{2}}} $$

Verify the derivatives and antiderivatives. $$ \frac{d}{d x} \ln \left(\frac{x-a}{x+a}\right)=\frac{2 a}{\left(x^{2}-a^{2}\right)} $$

Verify the derivatives and antiderivatives. $$ \frac{d}{d x} \ln \left(\frac{1+\sqrt{1-x^{2}}}{x}\right)=-\frac{1}{x \sqrt{1-x^{2}}} $$

Verify the derivatives and antiderivatives. $$ \frac{d}{d x} \ln \left(x+\sqrt{x^{2}-a^{2}}\right)=\frac{1}{\sqrt{x^{2}-a^{2}}} $$