# What is the Integration of coshx? | Integral of coshx

The integration of coshx is equal to sinhx+C. Here we learn how to integrate coshx, the cosine hyperbolic function.

The integration formula of cosh(x) is given as follows:

∫coshx dx = sinhx +C

where C denotes an integral constant.

## How to Integrate coshx

Question: What is the integral of coshx? That is,

Find ∫coshx dx.

By the definition of cosine hyperbolic functions, it is known that

coshx = $\dfrac{e^x+e^{-x}}{2}$

Integrating both sides, the integral of coshx will be equal to

∫coshx dx = $\int \dfrac{e^x+e^{-x}}{2}$ +C, where C is an arbitrary constant.

= $\dfrac{1}{2}$ ∫(ex + e-x) dx + C

= $\dfrac{1}{2}$ [∫ex dx + ∫e-x dx] + C

= $\dfrac{1}{2}$ [ex + (-e-x)] + C, because ∫emx dx = emx/m for any integer m.

= $\dfrac{e^x-e^{-x}}{2}$ + C

= sinh(x) +C, as we know that sinh(x) = (ex – e-x)/2.

Therefore, the integration of cosh(x) is equal to sinh(x)+C, that is, ∫cosh(x) dx = sinh(x) +C where C is an integral constant.