Primitive of x by Secant of a x
Jump to navigation
Jump to search
Theorem
\(\ds \int x \sec a x \rd x\) | \(=\) | \(\ds \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots + \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + \cdots} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C\) |
where $E_{2 n}$ is the $2 n$th Euler number.
Proof
\(\ds \int x \sec a x \rd x\) | \(=\) | \(\ds \frac 1 {a^2} \int \theta \sec \theta \rd \theta\) | Substitution of $a x \to \theta$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{ \paren {-1}^n E_{2 n} \theta^{2 n} } {\paren {2 n}!} \rd \theta\) | Power Series Expansion for Secant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n E_{2 n} } {\paren {2 n}!} \int \theta^{2 n + 1} \rd \theta\) | Power Series is Termwise Integrable within Radius of Convergence | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C\) | Substituting back $\theta \to ax$ |
$\blacksquare$
Also see
- Primitive of $x \sin a x$
- Primitive of $x \cos a x$
- Primitive of $x \tan a x$
- Primitive of $x \cot a x$
- Primitive of $x \csc a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sec a x$: $14.456$