# Integration of Square Root of a^2+x^2

In this tutorial we shall drive the integration of square root of a^2+x^2, and solve this integration with the help of integration by parts methods.

The integral of $\sqrt {{a^2} + {x^2}}$ is of the form

OR

This integral can be written as

Here first function is $\sqrt {{a^2} + {x^2}}$ and second function will be $1$

Using formula for integration by parts, we have

Equation (i) becomes using above formula, we have

Putting the given integral $I = \int {\sqrt {{a^2} + {x^2}} } dx$, we get

But using the relation ${\sinh ^{ - 1}}x = \ln \left( {x + \sqrt {{x^2} + 1} } \right)$ and we know that ${\sinh ^{ - 1}}\left( {\frac{x}{a}} \right) = \ln \left( {x + \sqrt {{x^2} + {a^2}} } \right)$ we have