What is the integral of cosh(x)?

The integral of cosh(x) is sinh(x) + C, where C is the constant of integration. This result can be understood through the definition and properties of hyperbolic functions.

The hyperbolic cosine function, cosh(x), is defined as:

[ \cosh(x) = \frac{e^x + e^{-x}}{2} ]

To find the integral, we can recall the derivative of sinh(x), which is given by:

[ \frac{d}{dx} \sinh(x) = \cosh(x) ]

This means that the function sinh(x) is the antiderivative of cosh(x). Therefore, when integrating cosh(x), we arrive at:

[ \int \cosh(x) , dx = \sinh(x) + C ]

where C is the constant of integration that arises from the indefinite integral. This relationship highlights the fact that the integral and derivative processes are inversely related for these hyperbolic functions.

The other options do not align with the properties of hyperbolic functions or their derivatives. For instance, integrating cos(x) gives a result involving sine, while e^x and tan(x) are unrelated to the integral of cos