Primitive of Reciprocal of Root of x squared plus Constant

Theorem

 * $\ds \int \frac {\d x} {\sqrt {x^2 + k} } = \ln \size {x + \sqrt {x^2 + k} } + C$

Positive Constant
Let $k > 0$.

Then $k = a^2$ for some $a \in \R$.

Hence from Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$: Logarithm Form:

from which the result follows.

Negative Constant
Let $k < 0$.

Then $k = -a^2$ for some $a \in \R$.

Hence from Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$: Logarithm Form:

from which the result follows.

Zero Constant
Let $k = 0$.

Then we have:

The result follows.