WordGPT

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

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

Explain it step by step in chatMake a short video of it
Graph of f(x) = sinh(x) with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.6-0.50.511.522.5f(x)
Graph of f(x) = sinh(x) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.54308063.

How to work it out

  1. 1

    We want an antiderivative of f(x) = sinh(x).

    ∫sinh⁡(x) dx\int \sinh\left(x\right)\,dx
  2. 2

    Method: standard integral.

  3. 3

    That gives the antiderivative; add the constant C because any constant disappears when you differentiate:

    ∫sinh⁡(x) dx=cosh⁡(x)+C\int \sinh\left(x\right)\,dx = \cosh\left(x\right) + C
  4. 4

    Check by differentiating the answer; you must get the original function back:

    ddx[cosh⁡(x)]=sinh⁡(x)\frac{d}{dx}\left[\cosh\left(x\right)\right] = \sinh\left(x\right)
  5. 5

    As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:

    ∫01sinh⁡(x) dx=[cosh⁡(x)]01≈0.54308063\int_{0}^{1} \sinh\left(x\right)\,dx = \Big[\cosh\left(x\right)\Big]_{0}^{1} \approx 0.54308063
LaTeX source of the formulas on this page
\[ \int \sinh\left(x\right)\,dx = \cosh\left(x\right) + C \]
\[ \int \sinh\left(x\right)\,dx \]
\[ \frac{d}{dx}\left[\cosh\left(x\right)\right] = \sinh\left(x\right) \]
\[ \int_{0}^{1} \sinh\left(x\right)\,dx = \Big[\cosh\left(x\right)\Big]_{0}^{1} \approx 0.54308063 \]

Definite integrals of the same function

IntervalValue
from 0 to 10.54308063
from 0 to 0.50.12762597

Questions

What is the integral of sinh(x)?

The integral of sinh⁡(x)\sinh\left(x\right) is cosh⁡(x)+C\cosh\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 sinh(x) from 0 to 1?

About 0.54308063.

More integrals

    Integral of sinh(x): cosh(x) + C (with steps) | WordGPT