Primitive of Reciprocal of Power of x squared plus a squared

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Theorem

$\ds \int \frac {\d x} {\paren {x^2 + a^2}^n} = \frac x {2 \paren {n - 1} a^2 \paren {x^2 + a^2}^{n - 1} } + \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {x^2 + a^2}^{n - 1} }$


Proof

Aiming for an expression in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

in order to use the technique of Integration by Parts, let:

\(\ds v\) \(=\) \(\ds x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d v} {\d x}\) \(=\) \(\ds 1\) Power Rule for Derivatives


Thus:

\(\ds u\) \(=\) \(\ds \frac 1 {\paren {x^2 + a^2}^{n - 1} }\)
\(\ds \) \(=\) \(\ds \paren {x^2 + a^2}^{-\paren {n - 1} }\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds -2 \paren {n - 1} x \paren {x^2 + a^2}^{-\paren {n - 1} - 1}\) Power Rule for Derivatives and Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds -2 \paren {n - 1} x \paren {x^2 + a^2}^{-n}\)
\(\ds \) \(=\) \(\ds \frac {-\paren {2 n - 2} x} {\paren {x^2 + a^2}^n}\)

Then:

\(\ds \int \frac {\d x} {\paren {x^2 + a^2}^{n - 1} }\) \(=\) \(\ds \frac 1 {\paren {x^2 + a^2}^{n - 1} } x - \int x \frac {-\paren {2 n - 2} x} {\paren {x^2 + a^2}^n} \rd x\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac x {\paren {x^2 + a^2}^{n - 1} } + \paren {2 n - 2} \int \frac {x^2} {\paren {x^2 + a^2}^n} \rd x\) simplifying
\(\ds \) \(=\) \(\ds \frac x {\paren {x^2 + a^2}^{n - 1} } + \paren {2 n - 2} \int \frac {x^2 + a^2 - a^2} {\paren {x^2 + a^2}^n} \rd x\)
\(\ds \) \(=\) \(\ds \frac x {\paren {x^2 + a^2}^{n - 1} } + \paren {2 n - 2} \int \frac {x^2 + a^2} {\paren {x^2 + a^2}^n} \rd x - \paren {2 n - 2} a^2 \int \frac {\d x} {\paren {x^2 + a^2}^n}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac x {\paren {x^2 + a^2}^{n - 1} } + \paren {2 n - 2} \int \frac {\d x} {\paren {x^2 + a^2}^{n - 1} } - \paren {2 n - 2} a^2 \int \frac {\d x} {\paren {x^2 + a^2}^n}\)
\(\ds \leadsto \ \ \) \(\ds \paren {1 - \paren {2 n - 2} } \int \frac {\d x} {\paren {x^2 + a^2}^{n - 1} }\) \(=\) \(\ds \frac x {\paren {x^2 + a^2}^{n - 1} } - \paren {2 n - 2} a^2 \int \frac {\d x} {\paren {x^2 + a^2}^n}\)
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {\paren {x^2 + a^2}^n}\) \(=\) \(\ds \frac x {2 \paren {n - 1} a^2 \paren {x^2 + a^2}^{n - 1} } + \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {x^2 + a^2}^{n - 1} }\)

$\blacksquare$


Sources

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