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.

Integral of sinhx

Table of Contents

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.

= $\dfrac{1}{2}$ ∫(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

= coshx +C, as we know that cosh x = (ex + 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 xIntegration of tan x
Integration of cot xDerivative of cosh(x)
Derivative of tanh(x)Derivative of sinh(x)

FAQs

Q1: What is the integration of sinhx?

Answer: The integration of sinhx is equal to coshx+C, that is, ∫sinhx dx = coshx +C.

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