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

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Theorem

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


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 Integrals
\(\ds \) \(=\) \(\ds \frac {-1} {a^2 x^2} - \frac 1 {a^4} \int \frac {\d x} x + \frac 1 {a^4} \int \frac {x \rd x} {x^2 + a^2} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-1} {a^2 x^2} - \frac 1 {a^4} \ln \size x + \frac 1 {a^4} \int \frac {x \rd x} {x^2 + a^2} + C\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac {-1} {a^2 x^2} - \frac 1 {a^4} \ln \size x + \frac 1 {a^4} \paren {\frac 1 2 \map \ln {x^2 + a^2} } + C\) Primitive of $\dfrac x {x^2 + a^2}$
\(\ds \) \(=\) \(\ds -\frac 1 {a^2 x^2} - \frac 1 {2 a^4} \map \ln {x^2} + \frac 1 {2 a^4} \map \ln {x^2 + a^2} + C\) 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

$\blacksquare$


Sources