Primitive of Reciprocal of a squared minus x squared squared

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Theorem

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

for $x^2 < a^2$.


Proof

\(\ds \int \frac {\d x} {\paren {a^2 - x^2}^2}\) \(=\) \(\ds \int \paren {\frac 1 {4 a^2} \paren {\dfrac 1 {\paren {a - x}^2} + \frac 1 {\paren {a + x}^2} + \frac 1 {a \paren {a + x} } + \frac 1 {a \paren {a - x} } } } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds \int \paren {\frac 1 {4 a^2} \paren {\frac 1 {\paren {a - x}^2} + \frac 1 {\paren {a + x}^2} + \frac 2 {\paren {a^2 - x^2} } } } \rd x\) Reciprocal of Difference of Squares as Sum of Reciprocals: $\dfrac 2 {a^2 - x^2} = \dfrac 1 {a \paren {a + x} } + \dfrac 1 {a \paren {a - x} }$
\(\ds \) \(=\) \(\ds \frac 1 {4 a^2} \paren {\int \frac {\d x} {\paren {a - x}^2} + \int \frac {\d x} {\paren {a + x}^2} } + \frac 1 {2 a^2} \int \frac {\d x} {a^2 - x^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 {4 a^2} \paren {\frac 1 {a - x} - \frac 1 {a + x} } + \frac 1 {2 a^2} \int \frac {\d x} {a^2 - x^2}\) Primitive of Function of $a x + b$ and Primitive of Power
\(\ds \) \(=\) \(\ds \frac x {2 a^2 \paren {a^2 - x^2} } + \frac 1 {2 a^2} \int \frac {\d x} {a^2 - x^2}\) Reciprocal of Difference of Squares as Difference of Reciprocals: $\dfrac x {a^2 - x^2} = \dfrac 1 {2 \paren {a + x} } - \dfrac 1 {2 \paren {a - x} }$
\(\ds \) \(=\) \(\ds \frac x {2 a^2 \paren {a^2 - x^2} } + \frac 1 {2 a^2} \paren {\dfrac 1 {2 a} \map \ln {\frac {a + x} {a - x} } } + C\) Primitive of $\dfrac 1 {a^2 - x^2}$

Hence the result.

$\blacksquare$


Also presented as

This result is also seen presented in the form:

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


Also see


Sources

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