Primitive of x over Cosine of a x
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Theorem
\(\ds \int \frac {x \rd x} {\cos a x}\) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 + \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} + \cdots} + C\) |
where $E_n$ denotes the $n$th Euler number.
Proof
\(\ds \int \frac {x \rd x}{\cos a x}\) | \(=\) | \(\ds \int x \sec a x \rd x\) | Definition of Secant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C\) | Primitive of $x \sec a x$ |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cos a x$: $14.376$