Primitive of Hyperbolic Tangent Function/Proof 2
Jump to navigation
Jump to search
Theorem
- $\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$
Proof
| \(\ds \int \tanh x \rd x\) | \(=\) | \(\ds -i \int \tan i x \rd x\) | Hyperbolic Tangent in terms of Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\int \tan i x \rd \paren {i x}\) | Primitive of Function of Constant Multiple | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \cmod {\cos i x} + C\) | Primitive of $\tan x$: Cosine Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \cmod {\cosh x} + C\) | Definition of Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \ln {\cosh x} + C\) | Graph of Hyperbolic Cosine Function: $\cosh x > 0$ for all $x$ |
$\blacksquare$