Primitive of Tangent of a x over x
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Theorem
- $\ds \int \frac {\tan a x} x \rd x = a x + \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + \cdots + C$
where $B_n$ denotes the $n$th Bernoulli number.
Proof
\(\ds \int \frac {\tan a x} x \rd x\) | \(=\) | \(\ds \int \frac 1 x \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n}!} \rd x\) | Power Series Expansion for Tangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} a^{2 n - 1} } {\paren {2 n}!} \int x^{2 n - 2} \rd x\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} a^{2 n - 1} } {\paren {2 n}!} \paren {\frac {x^{2 n - 1} } {2 n - 1} } + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n - 1} } {\paren {2 n - 1} \paren {2 n}!} + C\) |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tan a x$: $14.436$