Primitive of Secant of a x over x
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Theorem
- $\ds \int \frac {\sec a x} x \rd x = \ln \size x + \frac {\paren {a x}^2} 4 + \frac {5 \paren {a x}^4} {96} + \frac {61 \paren {a x}^6} {4320} + \cdots + \frac {\paren {-1}^n E_n \paren {a x}^{2 n} } {\paren {2 n} \paren {2 n}!} + \cdots + C$
where $E_n$ is the $n$th Euler number.
Proof
| \(\ds \int \frac {\sec a x} x \rd x\) | \(=\) | \(\ds \int \frac 1 x \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n} } {\paren {2 n}!} \rd x\) | Power Series Expansion for Secant Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {E_0} x \rd x + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n E_{2 n} a^{2 n} } {\paren {2 n}!} \int x^{2 n - 1} \rd x\) | Primitive of Constant Multiple of Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac 1 x \rd x + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n E_{2 n} a^{2 n} } {\paren {2 n}!} \paren {\frac {x^{2 n} } {2 n} } + C\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size x + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n} } {\paren {2 n} \paren {2 n}!} + C\) | Primitive of Reciprocal |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sec a x$: $14.457$