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.

Answer:

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.

= $\dfrac{1}{2}$ ∫(e^x + e^-x) dx + C

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

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

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

= sinh(x) +C, as we know that sinh(x) = (e^x – 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.

Also Read:

Integration of sinh(x)	Integration of tan x
Integration of cot x	Derivative of cosh(x)
Derivative of tanh(x)	Derivative of sinh(x)

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