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

Theorem

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

Proof

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

\(=\)

\(\ds \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \arsinh \frac x a + C\)

Primitive of $\sqrt {x^2 + a^2}$ in $\arsinh$ form

\(\ds \)

\(=\)

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

$\arsinh \dfrac x a$ in Logarithm Form

\(\ds \)

\(=\)

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

simplifying

\(\ds \)

\(=\)

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

subsuming $\dfrac {-a^2 \ln a} 2$ into the arbitrary constant

$\blacksquare$

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Integrals of Irrational Algebraic Functions: $3.3.41$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.189$
• 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $25$: Fundamental Integration Formulas: $26$.
• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $7$: Integrals
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals