# Definite integral involving sine and hyperbolic sine

• MHB
• MountEvariste
In summary, the formula for calculating a definite integral involving sine and hyperbolic sine is: ∫sin(x)sinh(x)dx = (sin(x)cosh(x) - cos(x)sinh(x))/2 + C. To solve this type of integral, you can use integration techniques such as substitution or integration by parts. The purpose of using a definite integral of this form is to find the area under the curve of the function sin(x)sinh(x) within a specific interval. Some common real-world applications include calculating work, displacement, and present value in physics, engineering, and financial mathematics. There can only be one solution for a definite integral involving sine and hyperbolic sine, but different approaches may be used
MountEvariste
Calculate $\displaystyle \int_0^{\infty} \frac{\sin x}{\cos x + \cosh x}\, \mathrm dx.$

If nothing else, you can express these functions as exponentials.

$sin(x)= \frac{e^{ix}- e^{-ix}}{2i}$
$cos(x)= \frac{e^{ix}+ e^{-ix}}{2}$
$cosh(x)= \frac{e^x+ e^{-x}}{2}$

Country Boy said:
If nothing else, you can express these functions as exponentials.

$sin(x)= \frac{e^{ix}- e^{-ix}}{2i}$
$cos(x)= \frac{e^{ix}+ e^{-ix}}{2}$
$cosh(x)= \frac{e^x+ e^{-x}}{2}$
Can you derive an infinite series of the integrand using these definitions?

Show that $\displaystyle \frac{\sin x }{\cos x + \cosh x} = i \bigg( \frac{1}{1+e^{ix-x}}-\frac{1}{1+e^{-ix-x}}\bigg)$.

Expand the RHS into geometric series to get, for $x \ge 0$:

$\displaystyle \frac{\sin x}{\cos x + \cosh x}=2\sum_{n=1}^{\infty}(-1)^{n-1}\sin(nx)e^{-nx}.$

Source: Yaghoub Sharifi.

Using the infinite sum in post #4, the answer follows by switching the order of sum and integral and using the result

$\displaystyle \int_0^{\infty} e^{-ax}\sin{bx} \, \mathrm dx =\frac{b}{a^2+b^2}$

Which can be derived via integration by parts or considering the fact that $\Im \left( e^{-ax+ibx} \right) = e^{-ax}\sin bx.$

See a detailed solution in this blog.

## 1. What is a definite integral involving sine and hyperbolic sine?

A definite integral involving sine and hyperbolic sine is a type of integral where the integrand contains both the sine function (sinx) and the hyperbolic sine function (sinh x). It is represented by the following formula: ∫absinx sinh x dx.

## 2. How do you solve a definite integral involving sine and hyperbolic sine?

To solve a definite integral involving sine and hyperbolic sine, you can use integration by parts or the substitution method. Integration by parts involves breaking down the integral into two parts and using the product rule to integrate them. The substitution method involves substituting a variable for one of the functions and then using the chain rule to integrate.

## 3. What is the purpose of using a definite integral involving sine and hyperbolic sine?

The purpose of using a definite integral involving sine and hyperbolic sine is to calculate the area under the curve of a function that contains both the sine function and the hyperbolic sine function. This can be useful in solving problems related to physics, engineering, and other fields.

## 4. Are there any special properties or rules for definite integrals involving sine and hyperbolic sine?

Yes, there are a few special properties and rules for definite integrals involving sine and hyperbolic sine. For example, the integral of the product of two functions is equal to the product of their integrals. Also, the integral of the sum of two functions is equal to the sum of their integrals. Additionally, there are specific rules for integrating trigonometric and hyperbolic functions, which can be applied to definite integrals involving sine and hyperbolic sine.

## 5. Can definite integrals involving sine and hyperbolic sine be solved using numerical methods?

Yes, definite integrals involving sine and hyperbolic sine can be solved using numerical methods such as the trapezoidal rule, Simpson's rule, and Gaussian quadrature. These methods involve approximating the integral by dividing the interval into smaller subintervals and using numerical techniques to calculate the area under the curve. These methods are useful when the integral cannot be solved analytically or when the integrand is too complex to integrate by hand.

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