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

Theorem

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

for $x^2 < a^2$.

Proof

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

\(=\)

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

Partial Fraction Expansion

\(\ds \)

\(=\)

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

Linear Combination of Primitives

\(\ds \)

\(=\)

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

Primitive of Reciprocal

\(\ds \)

\(=\)

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

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

\(\ds \)

\(=\)

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

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

\(\ds \)

\(=\)

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

Logarithm of Power and $x^2 > 0$

\(\ds \)

\(=\)

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

Difference of Logarithms

$\blacksquare$

Also see

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.174$