Primitive of x over Root of x squared plus a squared

Theorem

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

Proof

Let:

\(\ds z^2\)

\(=\)

\(\ds x^2 + a^2\)

\(\ds \leadsto \ \ \)

\(\ds 2 z \frac {\d z} {\d x}\)

\(=\)

\(\ds 2 x\)

Chain Rule for Derivatives, Power Rule for Derivatives

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \int \frac {z \rd z} {z}\)

Integration by Substitution

\(\ds \)

\(=\)

\(\ds \int \rd z\)

\(\ds \)

\(=\)

\(\ds z + C\)

Primitive of Constant

\(\ds \)

\(=\)

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

substituting for $z$

$\blacksquare$

Also see

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.183$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(9)$ Integrals Involving $\sqrt {x^2 + a^2}$: $17.9.2.$