Primitive of Reciprocal of x by Root of x squared plus a squared/Reciprocal Logarithm Form

Theorem

$\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = \frac 1 a \map \ln {\frac x {a + \sqrt {x^2 + a^2} } } + C$

Proof

\(\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} }\)

\(=\)

\(\ds -\frac 1 a \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C\)

Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: Logarithm form

\(\ds \)

\(=\)

\(\ds \frac 1 a \map \ln {\frac x {a + \sqrt {a^2 + x^2} } } + C\)

Logarithm of Reciprocal

$\blacksquare$

Also see

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

Sources

• 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $5$. Integrals: Integrals of Special Functions: $29$