Chapter A6, Problem 38E

To determine

To verify: the integration formula.

Explanation of Solution

Given:

  ${\displaystyle \int x}{\mathrm{sech}}^{-1}xdx=\frac{x^2}{2}{\mathrm{sech}}^{-1}x-\frac{1}{2}\sqrt{1-x^2}+C$

Proof:

To verify, ${\displaystyle \int x}{\mathrm{sech}}^{-1}xdx=\frac{x^2}{2}{\mathrm{sech}}^{-1}x-\frac{1}{2}\sqrt{1-x^2}+C$

Take, $y=\frac{x^2}{2}{\mathrm{sech}}^{-1}x-\frac{1}{2}\sqrt{1-x^2}+C$

Differentiate $y$ with respect to $x$

  $\Rightarrow \frac{dy}{dx}=x{\mathrm{sech}}^{-1}x+\frac{x^2}{2}\left(\frac{-1}{x\sqrt{1-x^2}}\right)+\frac{2x}{4\sqrt{1-x^2}}$

  $\Rightarrow \frac{dy}{dx}=x{\mathrm{sech}}^{-1}x$

Hence verified

Conclusion:

Hence the given integration is verified.