Primitive of Cosine of a x over x
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Theorem
\(\ds \int \frac {\cos a x \rd x} x\) | \(=\) | \(\ds \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {-1}^k \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size x - \frac {\paren {a x}^2} {2 \times 2!} + \frac {\paren {a x}^4} {4 \times 4!} - \frac {\paren {a x}^6} {6 \times 6!} - \cdots + C\) |
Proof
\(\ds \int \frac {\cos a x \rd x} x\) | \(=\) | \(\ds \int \frac 1 x \paren {\sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {a x}^{2 k} }{\paren {2 k}!} } \rd x\) | Definition of Real Cosine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k a^{2 k} } {\paren {2 k}!} \int \frac 1 x \paren {x^{2 k} } \rd x\) | Linear Combination of Primitives | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \int \frac {\rd x} x + \sum_{k \mathop = 1}^\infty \frac {\paren {-1}^k a^{2 k} } {\paren {2 k}!} \int \paren {x^{2 k - 1} } \rd x\) | extracting case for $k = 0$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d x} x + \sum_{k \mathop = 1}^\infty \frac {\paren {-1}^k a^{2 k} } {\paren {2 k}!} \frac {x^{2 k} } {2 k} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d x} x + \sum_{k \mathop \ge 1} \frac {\paren {-1}^k \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {-1}^k \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C\) | Primitive of Reciprocal |
The validity of $(1)$ follows from Cosine Function is Absolutely Convergent.
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cos a x$: $14.373$