Primitive of Reciprocal of Root of x squared plus Constant

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Theorem

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


Proof

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:

$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \map \ln {x + \sqrt {x^2 + a^2} } + C$


from which the result follows.

$\Box$


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:

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

for $0 < a < \size x$.


from which the result follows.

$\Box$


Zero Constant

Let $k = 0$.

Then we have:

\(\ds \int \frac {\d x} {\sqrt {x^2 + k} }\) \(=\) \(\ds \int \frac {\d x} {\sqrt {x^2} }\)
\(\ds \) \(=\) \(\ds \int \frac {\d x} x\)
\(\ds \) \(=\) \(\ds \ln \size x + C\)
\(\ds \) \(=\) \(\ds \ln \size x + \ln 2 + C'\) where $C' - C - \ln 2$
\(\ds \) \(=\) \(\ds \ln \size {2 x} + C'\) Sum of Logarithms
\(\ds \) \(=\) \(\ds \ln \size {x + x} + C'\)
\(\ds \) \(=\) \(\ds \ln \size {x + \sqrt {x^2 + 0} } + C'\)

The result follows.

$\blacksquare$


Sources