21. Antiderivatives, Areas and the FTC

b. Antiderivative Rules

4. Hyperbolic Trig Functions (Optional)

The tables on the previous pages listed the antiderivatives of the standard special functions. The tables on this page, list the properties of the hyperbolic trig functions and their inverses. These were defined in the PreCalculus chapter on Exponentials and Logarithms. The derivatives of the hyperbolic trig functions were found in the chapter on Derivative Rules and the derivatives of their inverses were found in the chapter on Inverse Functions.

Definitions and Derivatives

We first review the definitions and derivatives of the hyperbolic trig functions. Their derivatives can be derived by differentiating their exponential formulas as done in the chapter on Derivative Rules.

Hyperbolic Trigonometric Function

	Definition	Derivative
Hyperbolic Sine	\(\sinh x=\dfrac{e^x-e^{-x}}{2}\)	\(\dfrac{d}{dx}\sinh x=\cosh x\)
Hyperbolic Cosine	\(\cosh x=\dfrac{e^x+e^{-x}}{2}\)	\(\dfrac{d}{dx}\cosh x=\sinh x\)
Hyperbolic Tangent	\(\tanh x=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}=\dfrac{\sinh x}{\cosh x}\)	\(\dfrac{d}{dx}\tanh x=\mathrm{sech}^2 x\)
Hyperbolic Cotangent	\(\coth x=\dfrac{e^x+e^{-x}}{e^x-e^{-x}}=\dfrac{\cosh x}{\sinh x}\)	\(\dfrac{d}{dx}\coth x=-\mathrm{csch}^2 x\)
Hyperbolic Secant	\(\mathrm{sech}\,x=\dfrac{2}{e^x+e^{-x}}=\dfrac{1}{\cosh x}\)	\(\dfrac{d}{dx}\mathrm{sech}\,x=-\,\mathrm{sech}\,x\tanh x\)
Hyperbolic Cosecant	\(\mathrm{csch}\,x=\dfrac{2}{e^x-e^{-x}}=\dfrac{1}{\sinh x}\)	\(\dfrac{d}{dx}\mathrm{csch}\,x=-\,\mathrm{csch}\,x\coth x\)

Notice that the derivatives are identical to those for the ordinary circular trig functions except for a few minus sign differences.

---

We now look at the inverse hyperbolic functions. These were defined in the PreCalculus chapter on Exponentials and Logarithms. Since there are exponential definitions of the hyperbolic functions, it is not surprising there are logarithmic formulas for their inverses. Their derivatives can be derived by differentiating their logarithmic formulas as done in the chapter on Inverse Functions.

Inverse Hyperbolic Trigonometric Function

	Formula	Derivative
Inverse Hyperbolic Sine	\(\mathrm{arcsinh}\,x=\ln(x+\sqrt{x^2+1})\)	\(\dfrac{d}{dx}\mathrm{arcsinh}\,x=\dfrac{1}{\sqrt{x^2+1}}\)
Inverse Hyperbolic Cosine	\(\mathrm{arccosh}\,x=\ln(x+\sqrt{x^2-1})\)	\(\dfrac{d}{dx}\mathrm{arccosh}\,x=\dfrac{1}{\sqrt{x^2-1}}\)
Inverse Hyperbolic Tangent	\(\mathrm{arctanh}\,x=\dfrac{1}{2}\ln\left(\dfrac{1+x}{1-x}\right)\)	\(\dfrac{d}{dx}\mathrm{arctanh}\,x=\dfrac{1}{x^2-1}\)
Inverse Hyperbolic Cotangent	\(\mathrm{arccoth}\,x=\dfrac{1}{2}\ln\left(\dfrac{x+1}{x-1}\right)\)	\(\dfrac{d}{dx}\mathrm{arccoth}\,x=\dfrac{-1}{x^2-1}\)
Inverse Hyperbolic Secant	\(\mathrm{arcsech}\,x=\ln\left(\dfrac{1+\sqrt{1-x^2}}{x}\right)\)	\(\dfrac{d}{dx}\mathrm{arcsech}\,x=\dfrac{-1}{\sqrt{x^2(1-x^2)}}\)
Inverse Hyperbolic Cosecant	\(\mathrm{arccsch}\,x=\ln\left(\dfrac{1+\sqrt{1+x^2}}{x}\right)\)	\(\dfrac{d}{dx}\mathrm{arccsch}\,x=\dfrac{-1}{\sqrt{x^2(1+x^2)}}\)

Notice that the derivatives of the inverse hyperbolic trig functions are the same as those for the ordinary inverse trig functions except for numerous minus sign differences.

Antiderivatives

We are now ready for the antiderivatives. You can check each line of the table by differentiating the quantity on the right to see you get the quantity on the left.

Hyperbolic Trigonometric Functions

	If the Function is	then the Antiderivative is
Hyperbolic Sine	\(f(x)=\cosh x\)	\(F(x)=\sinh x+C\)
Hyperbolic Cosine	\(f(x)=\sinh x\)	\(F(x)=\cosh x+C\)
Hyperbolic Tangent\(^\text{1}\)	\(f(x)=\mathrm{sech}^2 x\)	\(F(x)=\tanh x+C\)
Hyperbolic Cotangent\(^\text{1}\)	\(f(x)=\mathrm{csch}^2 x\)	\(F(x)=-\coth x+C\)
Hyperbolic Secant\(^\text{1}\)	\(f(x)=\mathrm{sech}\,x\,\mathrm{tanh}\,x\)	\(F(x)=-\,\mathrm{sech}\,x+C\)
Hyperbolic Cosecant\(^\text{1}\)	\(f(x)=\mathrm{csch}\,x\,\mathrm{coth}\,x\)	\(F(x)=-\,\mathrm{csch}\,x+C\)

\(^\text{1}\) These are the functions whose antiderivatives are the hyperbolic trig functions: \(\tanh x\), \(\coth x\), \(\mathrm{sech}\,x\) and \(\mathrm{csch}\,x\). We will discuss the antiderivatives of these hyperbolic trig functions in Calculus 2 in the chapter on Trig Integrals.

Inverse Hyperbolic Trigonometric Functions\(^\text{2}\)

	If the Function is	then the Antiderivative is
Inverse Hyperbolic Sine	\(f(x)=\dfrac{1}{\sqrt{x^2+1}}\)	\(F(x)=\mathrm{arcsinh}\,x+C\)
Inverse Hyperbolic Cosine	\(f(x)=\dfrac{1}{\sqrt{x^2-1}}\)	\(F(x)=\mathrm{arccosh}\,x+C\)
Inverse Hyperbolic Tangent	\(f(x)=\dfrac{1}{x^2-1}\)	\(F(x)=\mathrm{arctanh}\,x+C\)
Inverse Hyperbolic Cotangent\(^\text{3}\)	\(f(x)=\dfrac{1}{x^2-1}\)	\(F(x)=-\,\mathrm{arccoth}\,x+C\)
Inverse Hyperbolic Secant	\(f(x)=\dfrac{1}{\sqrt{x^2(1-x^2)}}\)	\(F(x)=-\,\mathrm{arcsech}\,x+C\)
Inverse Hyperbolic Cosecant	\(f(x)=\dfrac{1}{\sqrt{x^2(1+x^2)}}\)	\(F(x)=-\,\mathrm{arccsch}\,x+C\)

\(^\text{2}\) These are the functions whose antiderivatives are the inverse hyperbolic trig functions. We will discuss the antiderivatives of the inverse hyperbolic trig functions in Calculus 2 in the chapter on Integration by Parts.

\(^\text{3}\) This antiderivative formula is not useful, since the formula using \(\mathrm{arctanh}\,x\) is easier. They just differ by a constant.