Integrating hyperbolic functions is similar to integrating their trigonometric counterparts, but there are some key differences due to the properties of hyperbolic functions. Here are the basic integrals for the six primary hyperbolic functions:
(1) Sinh (Hyperbolic Sine):
$$
\int \sinh(x) \, dx = \cosh(x) + C
$$
The integral of hyperbolic sine is hyperbolic cosine.
Proof
To integrate$\sinh(x)), we integrate this expression:
$$
\int \sinh(x) \, dx = \int \frac{e^x – e^{-x}}{2} \, dx
$$
$$
\int \sinh(x) \, dx = \frac{1}{2} \int e^x \, dx – \frac{1}{2} \int e^{-x} \, dx
$$
$$
\int \sinh(x) \, dx = \frac{1}{2} (e^x – (-e^{-x})) + C
$$
$$
\int \sinh(x) \, dx = \frac{1}{2} (e^x + e^{-x}) + C
$$
And this is the hyperbolic cosine function,$\cosh(x)). Therefore:
$$
\int \sinh(x) \, dx = \cosh(x) + C
$$
where$C) is the constant of integration.
(2) Cosh (Hyperbolic Cosine):
$$
\int \cosh(x) \, dx = \sinh(x) + C
$$
The integral of hyperbolic cosine is hyperbolic sine.
Proof
To integrate$\sinh(x)), we integrate this expression:
$$
\int \cosh(x) \, dx = \int \frac{e^x + e^{-x}}{2} \, dx
$$
$$
\int \cosh(x) \, dx = \frac{1}{2} \int e^x \, dx + \frac{1}{2} \int e^{-x} \, dx
$$
$$
\int \cosh(x) \, dx = \frac{1}{2} (e^x – e^{-x}) + C
$$
$$
\int \cosh(x) \, dx = \frac{1}{2} (e^x – e^{-x}) + C
$$
And this is the hyperbolic sin function,$\sinh(x)). Therefore:
$$
\int \cosh(x) \, dx = \sinh(x) + C
$$
where$C) is the constant of integration.
(3) Tanh (Hyperbolic Tangent):
$$
\int \tanh(x) \, dx = \ln(\cosh(x)) + C
$$
The integral of hyperbolic tangent involves the natural logarithm of hyperbolic cosine.
Proof
To integrate$\tanh(x)), we integrate this expression:
$$
\int \tanh(x) \, dx = \int \frac{sinh(x)}{cosh(x)} \, dx
$$
Let t= cosh(x)
dt= sinh(x) dx
$$
\int \tanh(x) \, dx = \int \frac {1}{t}\, dt= ln |t| + C
$$
And this is the hyperbolic tan function,$\tanh(x)). Therefore:
$$
\int \tan h(x) \, dx = ln | cosh(x)| + C
$$
where$C) is the constant of integration.
(4) Coth (Hyperbolic Cotangent):
$$
\int \coth(x) \, dx = \ln|\sinh(x)| + C
$$
The integral of hyperbolic cotangent involves the natural logarithm of the absolute value of hyperbolic sine.
Proof
To integrate$\coth(x)), we integrate this expression:
$$
\int \coth(x) \, dx = \int \frac{cosh(x)}{sinh(x)} \, dx
$$
Let t= sinh(x)
dt= cosh(x) dx
$$
\int \coth(x) \, dx = \int \frac {1}{t}\, dt= ln |t| + C
$$
And this is the hyperbolic cot function,$\coth(x)). Therefore:
$$
\int \cot h(x) \, dx = ln | sinh(x)| + C
$$
where$C) is the constant of integration.
(5) Sech (Hyperbolic Secant):
$$
\int sech(x) \, dx = \arctan(\sinh(x)) + C
$$
The integral of hyperbolic secant involves the inverse tangent of hyperbolic sine.
(6) Csch (Hyperbolic Cosecant):
$$
\int csch(x) \, dx = \ln|\tanh(x/2)| + C
$$
The integral of hyperbolic cosecant involves the natural logarithm of the absolute value of the hyperbolic tangent of half the angle.
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