What is the Integral of sinhx? | sinhx Integration

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

The sinh(x) integral formula is given below:

∫sinhx dx = coshx +C

where C is an integration constant.

How to Integrate sinhx

Question: Find the integral of sinhx, that is,

Find ∫sinhx dx.

Answer:

One know that

sinhx = $\dfrac{e^x-e^{-x}}{2}$

Integrating both sides, we get that

∫sinhx dx = $\int \dfrac{e^x-e^{-x}}{2}$ +C. Here C is a 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

= coshx +C, as we know that cosh x = (e^x + e^-x)/2.

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

Also Read:

Integration of sec 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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