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


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