Question

Solve the integral

$\int \frac{1}{\sqrt{x^2+9}}dx$

Short answer

$\sin h^{-1}\left(\frac{x}{3}\right)+c$

Step-by-step solution

Step 1. Given 

The given integral is$\int \frac{1}{\sqrt{x^2+9}}dx$.

Step 2. Simplifying the integral using identity

$$\begin{gathered} \mathrm{To}\mathrm{solve}\mathrm{the}\mathrm{integral},\mathrm{le}tx=3sinh\left(u\right). \\ \mathrm{Now},\mathrm{derivation}\mathrm{of}\mathrm{the}\mathrm{above}\mathrm{equation}\mathrm{is}\mathrm{solved}\mathrm{below}. \\ x=3sinh\left(u\right). \\ dx=cosh\left(u\right)du \\ \mathrm{Now},\mathrm{use}\mathrm{the}\mathrm{identity} \\ sin{h}^2\left(u\right)+1=cos{h}^2\left(u\right) \\ \int \frac{1}{\sqrt{x^2+9}}dx=\int \frac{1}{\sqrt{\left(3sinh\right(u\left){.}^2\right)+9}}dx \\ =\int \frac{1}{3\sqrt{cos{h}^2\left(u\right)}}dx \\ =\int \frac{1}{\cos h\left(u\right)} \end{gathered}$$

Step 3. Solving the integral

$$\begin{gathered} \int \frac{1}{\sqrt{x^2+9}}dx \\ =\int 1du \\ =u \\ =\sin h^{-1}\left(\frac{x}{3}\right)+c \end{gathered}$$