Primitive of Reciprocal of x by x squared plus a squared/Proof 2

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Theorem

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


Proof

\(\ds \int \frac {\d x} {x \paren {x^2 + a^2} }\) \(=\) \(\ds \int \frac {a^2 \rd x} {a^2 x \paren {x^2 + a^2} }\) multiplying top and bottom by $a^2$
\(\ds \) \(=\) \(\ds \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x \paren {x^2 + a^2} }\) adding and subtracting $x^2$
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\paren {x^2 + a^2} \rd x} {x \paren {x^2 + a^2} } - \frac 1 {a^2} \int \frac {x^2 \rd x} {x \paren {x^2 + a^2} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\d x} x - \frac 1 {a^2} \int \frac {x \rd x} {x^2 + a^2}\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \ln \size x - \frac 1 {a^2} \int \frac {x \rd x} {x^2 + a^2} + C\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \ln \size x - \frac 1 {a^2} \paren {\frac 1 2 \map \ln {x^2 + a^2} } + C\) Primitive of $\dfrac x {x^2 + a^2}$
\(\ds \) \(=\) \(\ds \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 + a^2} } + C\) Difference of Logarithms

$\blacksquare$