Primitive of x squared over x squared plus a squared

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Theorem

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

where $a$ is a non-zero constant.


Proof

\(\ds \int \frac {x^2 \rd x} {x^2 + a^2}\) \(=\) \(\ds \int \paren {1 - \frac {a^2} {x^2 + a^2} } \rd x\) long division
\(\ds \) \(=\) \(\ds \int \d x - a^2 \int \frac {\d x} {x^2 + a^2}\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds x - a^2 \int \frac {\d x} {x^2 + a^2} + C\) Primitive of Constant
\(\ds \) \(=\) \(\ds x - a^2 \frac 1 a \arctan {\frac x a} + C\) Primitive of $\dfrac 1 {x^2 + a^2}$
\(\ds \) \(=\) \(\ds x - a \arctan {\frac x a} + C\) simplifying

$\blacksquare$


Sources