Primitive of Reciprocal of x by x squared plus a squared

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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 1

\(\ds \int \frac {\d x} {x \paren {x^2 + a^2} }\) \(=\) \(\ds \int \paren {\frac 1 {a^2 x} - \frac x {a^2 \paren {x^2 + a^2} } } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\d x} x - \frac 1 {a^2} \int \frac {x \rd x} {x^2 + a^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \ln \size x - \frac 1 {a^2} \int \frac {x \rd x} {x^2 + a^2} + C\) 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

$\blacksquare$


Proof 2

\(\ds \int \frac {\d x} {x \paren {x^2 + a^2} }\) \(=\) \(\ds \int \frac {a^2 \rd x} {a^2 x \paren {x^2 + a^2} }\) 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

$\blacksquare$


Proof 3

From Primitive of Reciprocal of x by Power of x plus Power of a:

$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$


So:

\(\ds \int \frac {\d x} {x \paren {x^2 + a^2} }\) \(=\) \(\ds \frac 1 {2 a^2} \ln \size {\frac {x^2} {x^2 + a^2} } + C\) Primitive of $\dfrac 1 {x \paren {x^n + a^n} }$ with $n = 2$
\(\ds \) \(=\) \(\ds \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 + a^2} } + C\) Absolute Value of Even Power

$\blacksquare$


Sources

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