Primitive of Reciprocal of x cubed by x squared minus a squared

Theorem

$\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2} } = \frac 1 {2 a^2 x^2} + \frac 1 {2 a^4} \map \ln {\frac {x^2 - a^2} {x^2} } + C$

for $x^2 > a^2$.

Proof

\(\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2} }\)

\(=\)

\(\ds \int \paren {\frac {-1} {a^2 x^3} - \frac 1 {a^4 x} + \frac x {a^4 \paren {x^2 - a^2} } } \rd x\)

Partial Fraction Expansion

\(\ds \)

\(=\)

\(\ds \frac {-1} {a^2} \int \frac {\d x} {x^3} - \frac 1 {a^4} \int \frac {\d x} x + \frac 1 {a^4} \int \frac {x \rd x} {x^2 - a^2}\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \frac 1 {2 a^2 x^2} - \frac 1 {a^4} \int \frac {\d x} x + \frac 1 {a^4} \int \frac {x \rd x} {x^2 - a^2} + C\)

Primitive of Power

\(\ds \)

\(=\)

\(\ds \frac 1 {2 a^2 x^2} - \frac 1 {a^4} \ln \size x + \frac 1 {a^4} \int \frac {x \rd x} {x^2 - a^2} + C\)

Primitive of Reciprocal

\(\ds \)

\(=\)

\(\ds \frac 1 {2 a^2 x^2} - \frac 1 {a^4} \ln \size x + \frac 1 {a^4} \paren {\frac 1 2 \map \ln {x^2 -a^2} } + C\)

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

\(\ds \)

\(=\)

\(\ds \frac 1 {2 a^2 x^2} - \frac 1 {2 a^4} \map \ln {x^2} + \frac 1 {2 a^4} \map \ln {x^2 - a^2} + C\)

Logarithm of Power

\(\ds \)

\(=\)

\(\ds \frac 1 {2 a^2 x^2} + \frac 1 {2 a^4} \map \ln {\frac {x^2 - a^2} {x^2} } + C\)

Difference of Logarithms

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {x^3 \paren {a^2 - x^2} }$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.150$