What is the integral of sinh(x) dx?

The integral of sinh(x) dx is cosh(x) + C because of the relationship between the hyperbolic sine and hyperbolic cosine functions.

To understand why this is the case, recall the definitions of the hyperbolic functions:

• The hyperbolic sine function is defined as ( \sinh(x) = \frac{e^x - e^{-x}}{2} ).

• The hyperbolic cosine function is defined as ( \cosh(x) = \frac{e^x + e^{-x}}{2} ).

When you differentiate cosh(x), you find that ( \frac{d}{dx} \cosh(x) = \sinh(x) ). This means that the antiderivative, or the integral, of sinh(x) must be cosh(x) plus a constant of integration (C). Therefore, when performing the integral, you ascertain:

[

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

]

Thus, the answer is confirmed as cosh(x) + C, demonstrating a direct connection between differentiation and integration when dealing with these hyperbolic functions.