Primitive of Root of x squared plus a squared over x squared/Inverse Hyperbolic Sine Form

Theorem

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

Proof

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Also see

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

Sources

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $24$.