Primitive of Hyperbolic Cosine of a x over x

From ProofWiki
Jump to navigation Jump to search

Theorem

\(\ds \int \frac {\cosh a x \rd x} x\) \(=\) \(\ds \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C\)
\(\ds \) \(=\) \(\ds \ln \size x + \frac {\paren {a x}^2} {2 \times 2!} + \frac {\paren {a x}^4} {4 \times 4!} + \frac {\paren {a x}^6} {6 \times 6!} + \cdots + C\)


Proof

\(\ds \int \frac {\cosh a x \rd x} x\) \(=\) \(\ds \int \frac 1 x \paren {\sum_{k \mathop = 0}^\infty \frac {\paren {a x}^{2 k} } {\paren {2 k}!} } \rd x\) Power Series Expansion for Hyperbolic Cosine Function
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \frac {a^{2 k} } {\paren {2 k}!} \int \frac 1 x \paren {x^{2 k} } \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \frac {\d x} x + \sum_{k \mathop = 1}^\infty \frac {a^{2 k} } {\paren {2 k}!} \int \paren {x^{2 k - 1} } \rd x\) extracting case for $k = 0$
\(\ds \) \(=\) \(\ds \int \frac {\d x} x + \sum_{k \mathop = 1}^\infty \frac {a^{2 k} } {\paren {2 k}!} \frac {x^{2 k} } {2 k} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \int \frac {\d x} x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C\) simplifying
\(\ds \) \(=\) \(\ds \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C\) Primitive of Reciprocal

$\blacksquare$


Also see


Sources