Question

Differentiate $\sin h^{-1}\left(\frac{x}{3}\right)$ with respect to $x$. Find out the solution of the integration $\int \frac{1}{(x^2+9)}dx$ Further find out the value of the integral $\int \frac{1}{(x^2+49)}dx$ ?

Answer 1

$\frac{d}{dx}{\sinh }^{-1}x=\frac{1}{\sqrt{x^2+1}}$

$\Rightarrow \frac{d}{dx}{\sinh }^{-1}\left(\frac{x}{3}\right)=\frac{1}{\sqrt{{\left(\frac{x}{3}\right)}^2+1}}\times \left(\frac{1}{3}\right)$

$\Rightarrow \frac{d}{dx}{\sinh }^{-1}\left(\frac{x}{3}\right)=\frac{1}{\sqrt{x^2+9}}$

As integration is reverse path way of differentiation ;

$\int \frac{1}{\sqrt{x^2+9}}={\sinh }^{-1}\left(\frac{x}{3}\right)$

And $\int \frac{1}{x^2+9}=\frac{1}{3}{\tan }^{-1}\left(\frac{x}{3}\right)$

$\Rightarrow \int \frac{1}{x^2+49}=\frac{1}{7}{\tan }^{-1}\left(\frac{x}{7}\right)$

And $\Rightarrow \int \frac{1}{\sqrt{x^2+49}}={\sinh }^{-1}\left(\frac{x}{7}\right)$