Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Also presented as

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Primitive of $\frac 1 {\sqrt {x^2 + a^2} }$: Logarithm Form: Also presented as

The standard presentation of this result is:

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


Some sources present this in the form:

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

which is the same as above, except that the constant $a$ has not been subsumed into the arbitrary constant $C$.


Some sources present it as:

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


Sources

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