Primitive of Root of x squared plus a squared over x

Theorem

Logarithm Form

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

Inverse Hyperbolic Sine Form

$\ds \int \frac {\sqrt {x^2 + a^2} } x \rd x = \sqrt {x^2 + a^2} - a \arsinh \size {\dfrac a x} + C$

Also see

• Primitive of $\dfrac {\sqrt {x^2 - a^2} } x$
• Primitive of $\dfrac {\sqrt {a^2 - x^2} } x$