Introduction to integral of sinh 2x

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 calculator 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 sine. You will also understand how to compute sin's integral by using different integration techniques.

What is the integral of sinh(2x)?

The integral sinh 2x is an antiderivative of sinh function which is equal to coshx. It is also known as the reverse derivative of sine function which is a hyperbolic function. By definition, a hyperbolic function is a relation between two exponential functions e^x and e^-x. Mathematically, it is expressed as;

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

Integration of sinh2x formula

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

$∫\sinh 2xdx = \frac{\cosh 2x}{2}+c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. You can also calculate the integral of sinh x by using above formula.

How to calculate the integral of sinh(2x)?

The integral of sinh2x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of sine by using:

1. Derivatives
2. Substitution method
3. Definite integral

Integral of sinh 2x by using derivatives

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. Therefore, we can use the derivative to calculate the integral of a function. Let’s discuss calculating the integration of sinh 2x by using derivatives.

Proof of integral of sinh 2x by using derivatives

Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integral of sinh 2x by using its derivative. For this, we have to look for some derivatives formulas or a formula that gives sinh 2x as the derivative of any function.

In derivative, we know that,

$\frac{d}{dx}(\cosh 2x)=2\sinh 2x$

It means that the derivative of cosh 2x gives us sinh 2x. Now by using integral, the integral of sinh x is:

$∫\sinh 2xdx=\frac{\cosh 2x}{2}+c$

Hence the integral of sinh(2x) is equal to the cosh 2x/2.

Integral of sinh 2x by using substitution method

The u-substitution method calculator 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 sin by using the substitution method.

Proof of integral of sinh2x by using substitution method

To proof the integration of sinh2x by using substitution method, suppose that:

$y = \sinh 2x$

Differentiating with respect to x,

$\frac{dy}{dx}=2\cosh 2x$

To calculate integral, we can write the above equation as:

$dy=\cosh 2x dx$

By trigonometric identities, we know that cosh 2x = √1 + sinh²2x. Then the above equation becomes,

$dy=\sqrt{1+\sinh^2 2x}.dx$

Now, substituting the value of 2sinh2 x, such as:

$dy=\sqrt{1+y^2}.dx$

Multiplying both sides by sinh2x,

$\frac{\sinh 2xdy}{\sqrt{1+y^2}}= \sinh2x dx$

Again substitute sinh2x = y on the left side.

$\frac{ydy}{\sqrt{1+y^2}}=\sinh 2xdx$

Integrating on both sides by applying integral,

$∫\frac{ydy}{\sqrt{1+y^2}}=∫\sinh 2xdx$

Let 1+ y² = u. Then 2y dy = du (or) y dy = 1/2 du.

Then the above left-hand side integral becomes,

$\frac{1}{2}∫\frac{1}{\sqrt u}du=∫\sinh 2xdx$

$\frac{1}{2}∫u^{-\frac{1}{2}}du=∫\sinh 2xdx$

Since the power rule of integration is

$∫x^ndx=\frac{x^{n+1}}{n+1}+C$

Therefore, by using this formula we get,

$\frac{1}{2}\left(\frac{u1/2}{1/2}\right)+C=∫\sinh 2xdx$

$u^{\frac{1}{2}} + C = ∫\sinh 2x.dx$

Again substituting u = 1 + y², we get

$\frac{1}{2}(1 + y^2)^{\frac{1}{2}}+C=∫\sinh 2x.dx$

And again Substitute y = sinh 2x here,

$\frac{1}{2}(1 + \sinh^2 2x)^{\frac{1}{2}}+C=∫\sinh 2xdx$

$\frac{1}{2}(\cosh^2 2x)^{\frac{1}{2}}+ C=∫\sinh 2xdx$

$\frac{\cosh 2x}{2}+C=∫\sinh 2x.dx$

Hence the integral of sinh 2x is -cosh 2x/2. Also use our trigonometric calculator to integral integrals by using trigonometric substitution. 

Integral of sinh2x 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 integration of sinh 2x by using the indefinite integral.

Proof of integral of sinh 2x by using definite integral

To compute the integral of sinh2x by using a definite integration calculator, we can use the interval from 0 to π or 0 to π/2. Let’s compute the integral of sinh(2x) from 0 to π. For this we can write the integral as:

$∫^π_0 \sinh 2x.dx = \left|\frac{\cosh 2x}{2}\right|^π_0 $

Now, substituting the limit in the given function.

$∫^π_0 \sinh 2x.dx=\frac{\cosh (2π)}{2}- \frac{\cosh2(0)}{2}$

Since cos 0 is equal to 1 and cos π is equal to -1, therefore,

$∫^π_0 \sinh 2x.dx = \frac{1}{2}–\frac{1}{2}=0$

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

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

Now,

$∫^{\frac{π}{2}}_0 \sinh2x dx=\frac{\cosh (π/2)}{2}-\frac{\cosh (0)}{2}$

Since cos 0 is equal to 1 and cos π/2 is equal to 0, therefore,

$∫^{\frac{π}{2}}_0 \sinh 2x dx = 0 – \frac{1}{2}= -\frac{1}{2}$

Therefore, the definite integration of sinh2x is equal to -1/2.