Antiderivatives of Hyperbolic Functions

The definitions corresponding to the hyperbolic functions was already given in the unit corresponding to the justification of the formulas for their derivatives. From those rules, the corresponding formulas to compute their antiderivatives can be readily deduced.

In this unit some other basic formulas to compute antiderivatives of hyperbolic functions are justified.

Antiderivative of the hyperbolic tangent function

By definition,

$\begin{equation*} \tanh x = \frac{\sinh x}{\cosh x} \end{equation*}$

To compute the antiderivative of this function, apply a change of variable setting $v = \cosh x$ so that its differential $dv = \sinh x\cdot dx$ appears at the numerator of the fraction. With this definition, the formula $\int dv / v$ is applicable:

$\begin{eqnarray*} \int\!\tanh x\cdot dx &=& \int\! \frac{\sinh x}{\cosh x} \cdot dx \\ &=& \int\!\frac{dv}{v} = \ln| v | + \hat{C} \\ &=& \ln|\cosh x| + C \end{eqnarray*}$

The application of the chain rule leads to the general rule for this antiderivative:

$\begin{equation*} \int\!\tanh v\cdot dv = \ln|\cosh v| + C \end{equation*}$

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Antiderivatives of Hyperbolic Functions

The formulas to compute the antiderivative of some hyperbolic functions are deduced.

Antiderivative of the hyperbolic tangent function