Primitive of Sine of a x over x
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Theorem
| \(\ds \int \frac {\sin a x \d x} x\) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \frac {\paren {-1}^k \paren {a x}^{2 k + 1} } {\paren {2 k + 1} \paren {2 k + 1}!} + C\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds a x - \frac {\paren {a x}^3} {3 \times 3!} + \frac {\paren {a x}^5} {5 \times 5!} - \cdots\) |
Proof
| \(\ds \int \frac {\sin a x \rd x} x\) | \(=\) | \(\ds \int \frac 1 x \paren {\sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {a x}^{2 k + 1} } {\paren {2 k + 1}!} } \rd x\) | Definition of Real Sine Function | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k a^{2 k + 1} } {\paren {2 k + 1}!} \int \frac 1 x \paren {x^{2 k + 1} } \rd x\) | Linear Combination of Primitives | ||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k a^{2 k + 1} } {\paren {2 k + 1}!} \int x^{2 k} \rd x\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k a^{2 k + 1} } {\paren {2 k + 1}!} \frac {x^{2 k + 1} } {2 k + 1} + C\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \frac {\paren {-1}^k \paren {a x}^{2 k + 1} } {\paren {2 k + 1} \paren {2 k + 1}!} + C\) | simplifying |
The validity of $(1)$ follows from Sine Function is Absolutely Convergent.
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.343$