Primitive of x over Power of x squared plus a squared

Theorem

$\ds \int \frac {x \rd x} {\paren {x^2 + a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 + a^2}^{n - 1} }$

Let:

$\ds z$

$=$

$\ds x^2 + a^2$

$\ds \leadsto \ \ $

$\ds \frac {\d z} {\d x}$

$=$

$\ds 2 x$

$\ds \leadsto \ \ $

$\ds \int \frac {x \rd x} {\paren {x^2 + a^2}^n}$

$=$

$\ds \int \frac {\d z} {2 z^n}$

$\ds $

$=$

$\ds \frac 1 2 \int z^{-n} \rd z$

$\ds $

$=$

$\ds \frac 1 2 \frac {z^{-n + 1} } {-n + 1} + C$

$\ds $

$=$

$\ds \frac {-1} {2 \paren {n - 1} z^{n - 1} } + C$

simplifying

$\ds $

$=$

$\ds \frac {-1} {2 \paren {n - 1} \paren {x^2 + a^2}^{n - 1} } + C$

substituting for $z$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^2 + a^2$: $14.140$
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: $(6)$ Integrals Involving $x^2 + a^2$: $17.6.16.$