Primitive of x cubed over x squared plus a squared

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Theorem

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


Proof

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

$\blacksquare$


Sources

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