Integral of $\cosh\left(x\right)$

Q: What is the integral of cosh(x)?

The integral of ≤ft(x) is ≤ft(x) + C.

The integral of $\cosh\left(x\right)$ is $\sinh\left(x\right) + C$ .

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

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Graph of f(x) = cosh(x) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 1.1752012.

How to work it out

1
We want an antiderivative of f(x) = cosh(x).
$\int \cosh\left(x\right)\,dx$
2
Method: standard integral.
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int \cosh\left(x\right)\,dx = \sinh\left(x\right) + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\sinh\left(x\right)\right] = \cosh\left(x\right)$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} \cosh\left(x\right)\,dx = \Big[\sinh\left(x\right)\Big]_{0}^{1} \approx 1.1752012$

LaTeX source of the formulas on this page

\[ \int \cosh\left(x\right)\,dx = \sinh\left(x\right) + C \]
\[ \int \cosh\left(x\right)\,dx \]
\[ \frac{d}{dx}\left[\sinh\left(x\right)\right] = \cosh\left(x\right) \]
\[ \int_{0}^{1} \cosh\left(x\right)\,dx = \Big[\sinh\left(x\right)\Big]_{0}^{1} \approx 1.1752012 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	1.1752012
from 0 to 0.5	0.52109531

Questions

What is the integral of cosh(x)?

The integral of $\cosh\left(x\right)$ is $\sinh\left(x\right) + C$ .

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of cosh(x) from 0 to 1?

About 1.1752012.

Integral of cosh⁡(x)\cosh\left(x\right)cosh(x)

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $\cosh\left(x\right)$