Primitive of Arctangent of x over a over x

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Theorem

\(\displaystyle \int \frac 1 x \arctan \paren {\frac x a} \rd x\) \(=\) \(\displaystyle \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {x^{2 k + 1} } {\paren {2 k + 1}^2 a^{2 k + 1} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac x a - \frac {x^3} {3^2 a^3} + \frac {x^5} {5^2 a^5} - \frac {x^7} {7^2 a^7} + \cdots + C\)


Proof

\(\displaystyle \int \frac 1 x \arctan \paren {\frac x a} \rd x\) \(=\) \(\displaystyle \int \frac 1 x \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac 1 {2 k + 1} \paren {\frac x a}^{2 k + 1} \rd x\) Power Series Expansion for Real Arctangent Function
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac 1 {\paren {2 k + 1} a^{2 k + 1} } \int x^{2 k} \rd x\) Fubini's Theorem
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac{x^{2 k + 1} }{\paren {2 k + 1}^2 a^{2 k + 1} } + C\)


$\blacksquare$

Also see


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