Introduction to integral of cosh2x

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 trigonometric function sine. You will also understand how to compute cosh(2x) integral by using different integration techniques.

What is the integration of cosh2x?

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

$\cosh 2x=\frac{e^{2x}+e^{-2x}}{2}{2}nbsp;

The integral of cosh(2x) is a common integral in calculus. It contains a hyperbolic function cos with twice of angle x. It is used to solve many integral problems such as to solve the integral of cosh x.

Integral of cosh2x formula

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

$\int \cosh (2x)dx =\frac{\sinh(2x)}{2}+ c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. In the above formula, replacing cosh 2x by cos 2x gives the integral of cos(2x).

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

The integral of cosh 2x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the integral of cosine by using:

1. Derivatives
2. Substitution method
3. Definite integral

Integral of cosh 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 iintegral of cosh2x by using derivatives.

Proof of integral of cosh 2x by using derivatives

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

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

It means that the derivative of cosh(2x) gives us cosh x. But it has a negative sign. Therefore, to obtain the integral of cosh, we have to use it as the integral of cos, that is:

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

Hence the integral of cos x is equal to the negative of cosh x. It is written as:

$\int \cosh(2x)dx =\frac{\sinh(2x)}{2} + c$

Integral of cosh 2x by using substitution method

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

Proof of Integral of cosh 2x by using substitution method

To prove the integration of cosh2x by using the substitution method, suppose that:

$y = \cosh(2x)$

Differentiating with respect to x,

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

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

$dy = 2\sinh(2x).dx$

By trigonometric identities, we know that sinh2x = √cosh²2x – 1. Then the above equation becomes,

$dy=2\sqrt {\cosh^2(2x)-1} .dx$

Multiplying both sides by cos x,

$\frac{\cosh(2x)dy}{2\sqrt {\cosh^2(2x)-1}} = \cosh(2x).dx$

Again substitute cosh 2x = y on the left side.

$\frac{ydy}{2\sqrt {\cosh^2(2x)-1}} = \cosh(2x).dx$

Integrating on both sides by applying integral,

$\frac{ydy}{2\sqrt {\cosh^2(2x)-1}} =\int \cosh (2x)dx$

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

Then the above left-hand side integral becomes,

$\frac{1}{2}\int \frac{1}{2\sqrt u} du =\int \cosh(2x)dx$

$\frac{1}{2}\int \frac{u^{\frac{-1}{2}}}{2}du=\int \cosh(2x)dx$

Since the power rule of integration is ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C. Therefore, by using this formula we get,

$\frac{1}{2}\left(\frac{u^{1/2}}{2(1/2)}\right) + C =\int \cosh (2x)dx$

$\frac{u^{1/2}}{2}+ C =\int \cosh(2x)dx$

Again by using the u-substitution formula calculator, substitute u = 1 - y², we get

$\frac{(1 - y^2)^{1/2}}{2} + C = \int \cosh (2x) dx$

And again Substitute y = cosh 2x here,

$\frac{(1 - \cosh^22x)^{1/2}}{2}+ C =\int \cosh 2x dx$

$\frac{(\sinh^22x)^{1/2}}{2}+ C =\int \cosh 2x dx$

$\frac{\sinh 2x}{2} + C =\int \cosh 2x dx$

Hence the integral of cosh2x is sinh 2x/2.

Integral of cosh 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:

$\int^b_af(x) dx = F(b) – F(a)$

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

Proof of integral of cosh 2x by using definite integral

To compute the integral of cosh 2x by using a definite integral, we can use the interval from 0 to π or 0 to π/4. Let’s compute the integral of cosh 2x from 0 to π. For this, we can write the integral as:

$\int^\pi_0 \cosh(2x)dx = \left|\frac{\sinh 2x}{2}\right|^\pi_0$

Now, substitute the limit in the given function.

$\int^\pi_0 \cosh(2x)dx = \frac{\sinh 2(π)}{2} - \frac{\sinh 2(0)}{2}$

Since sinh 0 is equal to 0 and sinh π is equal to 0, therefore,

$\int^\pi_0 \cosh(2x)dx = 0$

Which is the calculation of the definite integral of cosh 2x. Now to calculate the integration of cosh2x between the interval 0 to π/4, we just have to replace π by π/4. Therefore,

$\int^{\frac{\pi}{4}}_0 \cosh(2x)dx = \left|\sinh 2x\right|^{\frac{\pi}{4}}_0$

Now,

$\int^{\frac{\pi}{4}}_0 \cosh(2x)dx = \frac{\sin 2(π/4)}{2} - \frac{sin 2(0)}{2}$

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

$\int^{\frac{\pi}{4}}_0 \cosh(2x)dx=\frac{1}{2}$

Therefore, the definite integral of cosh 2x is equal to 1/2. You can use our definite integration calculator to avoid long-term calculations.