Primitive of x cubed over x squared minus a squared squared

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Theorem

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

for $x^2 > a^2$.


Proof

\(\ds \int \frac {x^3 \rd x} {\paren {x^2 - a^2}^2}\) \(=\) \(\ds \int \frac {x \paren {x^2 - a^2 + a^2} } {\paren {x^2 - a^2}^2} \rd x\)
\(\ds \) \(=\) \(\ds \int \frac {x \paren {x^2 - a^2} } {\paren {x^2 - a^2}^2} \rd x + a^2 \int \frac {x \rd x} {\paren {x^2 - a^2}^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \frac {x \rd x} {x^2 - a^2} + a^2 \int \frac {x \rd x} {\paren {x^2 - a^2}^2}\) simplification
\(\ds \) \(=\) \(\ds \frac 1 2 \map \ln {x^2 - a^2} + a^2 \int \frac {x \rd x} {\paren {x^2 - a^2}^2} + C\) Primitive of $\dfrac x {x^2 - a^2}$
\(\ds \) \(=\) \(\ds \frac 1 2 \map \ln {x^2 - a^2} + a^2 \paren {-\frac 1 {2 \paren {x^2 - a^2} } } + C\) Primitive of $\dfrac x {\paren {x^2 - a^2}^2}$
\(\ds \) \(=\) \(\ds \frac {-a^2} {2 \paren {x^2 - a^2} } + \frac 1 2 \map \ln {x^2 - a^2} + C\) simplifying

$\blacksquare$


Also see


Sources

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