Primitive of Reciprocal of x squared minus a squared squared

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Theorem

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

for $x^2 > a^2$.


Proof

\(\ds \int \frac {\d x} {\paren {x^2 - a^2}^2}\) \(=\) \(\ds \int \paren {\frac 1 {4 a^3 \paren {x + a} } - \frac 1 {4 a^3 \paren {x - a} } + \frac 1 {4 a^2 \paren {x + a}^2} + \frac 1 {4 a^2 \paren {x - a}^2} } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3} \int \frac {\d x} {x + a} - \frac 1 {4 a^3} \int \frac {\d x} {x - a} + \frac 1 {4 a^2} \int \frac {\d x} {\paren {x + a}^2} + \frac 1 {4 a^2} \int \frac {\d x} {\paren {x - a}^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3} \ln \size {x + a} - \frac 1 {4 a^3} \ln \size {x - a} + \frac 1 {4 a^2} \int \frac {\d x} {\paren {x + a}^2} + \frac 1 {4 a^2} \int \frac {\d x} {\paren {x - a}^2} + C\) Primitive of Function of $a x + b$ and Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3} \ln \size {\frac {x + a} {x - a} } + \frac 1 {4 a^2} \int \frac {\d x} {\paren {x + a}^2} + \frac 1 {4 a^2} \int \frac {\d x} {\paren {x - a}^2} + C\) Difference of Logarithms
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3} \map \ln {\frac {x + a} {x - a} } + \frac 1 {4 a^2} \int \frac {\d x} {\paren {x + a}^2} + \frac 1 {4 a^2} \int \frac {\d x} {\paren {x - a}^2} + C\) Sign of Quotient of Factors of Difference of Squares
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3} \map \ln {\frac {x + a} {x - a} } + \frac 1 {4 a^2} \frac {-1} {\paren {x + a} } + \frac 1 {4 a^2} \frac {-1} {\paren {x - a} } + C\) Primitive of Function of $a x + b$ and Primitive of Power
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3} \map \ln {\frac {x + a} {x - a} } - \paren {\frac 1 {4 a^2 \paren {x + a} } + \frac 1 {4 a^2 \paren {x - a} } } + C\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3} \map \ln {\frac {x + a} {x - a} } - \paren {\frac {\paren {x - a} + \paren {x + a} } {4 a^2 \paren {x + a} \paren {x - a} } } + C\) simplifying
\(\ds \) \(=\) \(\ds \frac {-x} {2 a^2 \paren {x^2 - a^2} } + \frac 1 {4 a^3} \map \ln {\frac {x + a} {x - a} } + C\) Difference of Two Squares

$\blacksquare$


Also see


Sources

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