Primitive of Arctangent of x over a over x
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Theorem
\(\ds \int \frac 1 x \map \arctan {\frac x a} \rd x\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {x^{2 k + 1} } {\paren {2 k + 1}^2 a^{2 k + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac x a - \frac {x^3} {3^2 a^3} + \frac {x^5} {5^2 a^5} - \frac {x^7} {7^2 a^7} + \cdots + C\) |
Proof
\(\ds \int \frac 1 x \map \arctan {\frac x a} \rd x\) | \(=\) | \(\ds \int \frac 1 x \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac 1 {2 k + 1} \paren {\frac x a}^{2 k + 1} \rd x\) | Power Series Expansion for Real Arctangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac 1 {\paren {2 k + 1} a^{2 k + 1} } \int x^{2 k} \rd x\) | Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac{x^{2 k + 1} } {\paren {2 k + 1}^2 a^{2 k + 1} } + C\) |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.486$