Primitive of Square of Hyperbolic Tangent Function
Jump to navigation
Jump to search
Theorem
- $\ds \int \tanh^2 x \rd x = x - \tanh x + C$
where $C$ is an arbitrary constant.
Proof
| \(\ds \int \tanh^2 x \rd x\) | \(=\) | \(\ds \int \paren {1 - \sech^2 x} \rd x\) | Sum of Squares of Hyperbolic Secant and Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int 1 \rd x - \int \sech^2 x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int 1 \rd x - \tanh x + C\) | Primitive of Square of Hyperbolic Secant Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds x - \tanh x + C\) | Primitive of Constant |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.33$