Introduction to the integral of sinh^2(x)

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.

Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a hyperbolic function sinh squared. You will also understand how to compute the sinh square integral by using different integration techniques.

What is the integral of sinh^2x?

The integral of sinh^2x is an antiderivative of the sine function which is equal to x/2 - sinh(2x)/4. It is also known as the reverse derivative of the sine function which is a hyperbolic function. By definition, the hyperbolic function sine is a combination of two exponential functions e^x and e^-x. Mathematically, it can be expressed as:

$\sinh x=\frac{e^x-e^{-x}}{2}$

Integral of sinh²x formula

The formula of the integral of sinh contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(sinh²x)dx. In mathematical form, the integral of sin^2x is:

$∫\sinh^2xdx = \frac{x}{2}-\frac{\sinh2x}{4} + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. The above formula is also used to calcualte the integral of sinh x. 

How to calculate the integral of sinh^2?

The integral of sinh square x formula is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the integral of sine by using:

1. Integration by parts
2. Substitution method
3. Definite integral

Integral of sinh x squared by using integration by parts

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The by parts integration calculator is a method of solving integral of two functions combined together. Let’s discuss calculating the integral of sinh squared x by using integration by parts.

Proof of integral of sinh^2x by using integration by parts

Since we know that the function sine squared x can be written as the product of two functions. Therefore, we can calculate the integral of sinh^2(x) by using integration by parts. For this, suppose that:

$I = \sinh x.\sinh x$

Applying the integral we get,

$I = ∫\sinh x.\sinh xdx$

Since the method of integration by parts is:

$∫[f(x).g(x)]dx = f(x).∫g(x)dx - ∫[f’(x).∫g(x)dx]dx$

Now replacing f(x) and g(x) by sin x, we get,

$I=-\sinh x.\cosh x + ∫[\cosh x.\cosh x]dx$

It can be written as:

$I=-\sinh x.\cosh x + ∫[\cosh^2x]dx$

Now by using a trigonometric identity cosh²x = 1+cosh2x/2. Therefore, substituting the value of cosh²x in the above equation, we get:

$I=-\sinh x.\cosh x+∫\left(\frac{1+\cosh2x}{2}\right)dx$

Integrating remaining terms,

$I=-\sinh x.\cosh x +\frac{x}{2} + \frac{\sin 2x}{4}$

Or,

$I = -\frac{\sinh 2x}{2}+\frac{x}{2}+\frac{\sinh 2x}{4}$

Or,

$I = \frac{x}{2}–\frac{\sinh 2x}{4}$

Hence the integral of sin^2x is equal to,

$∫\sinh^2xdx=\frac{x}{2}–\frac{\sinh 2x}{4}{2}nbsp;

This method is also applicable to integrate sinh(2x).

Integral of sin^2x by using substitution method

The substitution method involves many trigonometric formulas. We can use these formulas to verify the integrals of different trigonometric functions such as sine, cosine, tangent, etc. Let’s understand how to prove the integral of sinh squared by using the substitution method.

Proof of integral of sinh square x formula by using substitution method

To prove the integral of sinh^2x by using substitution method, suppose that:

$I = ∫\sinh^2x = ∫(1- \cosh^2x)dx$

Further we can cos2x can be substituted as cosh²x =1+cosh(2x)/2. Then the above equation will become.

$I = x - ∫\left(\frac{1+ \cos2x}{2}\right)dx$

Integrating,

$I=x –\frac{x}{2}-\frac{\sin2x}{4}$

Moreover,

$I =\frac{x}{2}-\frac{\sin2x}{4}$

Hence the integral of sinh^2(x) is verified by using the substitution method. Also use our u-substitution method calculator to verify the above calculations. 

Integral of sin^2x by using definite integral

The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:

$∫^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integral of sin^2x by using the definite integral.

Proof of integral of sin^2x by using definite integral

To compute the integral of sin^2x by using a definite integral, we can use the interval from 0 to 2π or 0 to π. Let’s compute the integral of sinh^2 from 0 to 2π. The definite integral can be written as:

$∫^{2π}_0 \sinh^2x dx=\left|\frac{x}{2}-\frac{\sin 2x}{4}\right|^{2π}_0$

Substituting the value of limit we get,

$∫^{2π}_0 \sinh^2x dx=\left[\frac{2π}{2}-\frac{\sin 4π}{4}\right] -\left[0 - \frac{\sin 0}{4}\right]$

$∫^{2π}_0 \sinh^2x dx = π - \frac{0}{4}$

Therefore, the integral of sin2x from 0 to 2π is

$∫^{2π}_0 \sinh^2x dx = π$

Which is the calculation of the definite integral of sinh^2x. Now to calculate the integral of sinh ax between the interval 0 to π, we just have to replace π by π. Therefore,

$∫^π_0 \sinh^2x dx = \left|\frac{x}{2}-\frac{\sin 2x}{4}\right|^π_04

$∫^π_0 \sinh^2xdx=\left[\frac{π}{2}-\frac{\sin π}{4}\right] - \left[0 - \frac{\sin 0}{4}\right]$

$∫^π_0 \sinh^2x dx = \frac{π}{2} - \frac{0}{4}$

$∫^π_0 \sinh^2x dx = \frac{π}{2}$

Therefore, the integral of sin2x from 0 to π is π/2.