Introduction to integral of sinh 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 sinhx. You will also understand how to compute sin's integral by using different integration techniques.

What is the integral of sinhx?

The integral of sinhx is an antiderivative of the sine 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.

Integral of sinhx formula

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

$∫\sinh x dx = \cosh x + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. Different integration problems can be solved by using the above formula. 

How to calculate the integral of sinh(x)?

The integral of sinhx 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 x 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 integral of sinhx by using derivatives.

Proof of integral of sinh x by using derivatives

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

In derivative, we know that,

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

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

$∫\sinh xdx = \cosh x + c$

Hence the integral of sinhx is equal to the cosh x.

Integral of sinhx 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 sin by using the substitution method.

Proof of Integral of sinhx by using substitution method

There are two types of substitution methods in integrals, u-substitution, and trigonometric substitution. To prove the integral of sinhx by using the substitution method, suppose that:

$y = \sinh x$

Differentiating with respect to x,

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

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

$dy = \cosh x dx$

By trigonometric identities, we know that cosh x = √1 + sinh²x. Then the above equation becomes,

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

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

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

Multiplying both sides by sinhx,

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

Again substitute sinhx = y on the left side.

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

Integrating on both sides by applying integral,

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

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

Then the above left-hand side integral becomes,

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

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

Since the power rule of integration is

$∫x^ndx=\frac{x^{n+1}}{n+1}+C$

Therefore, by using this formula we get,

$\frac{1}{2}\left(\frac{u^{fac{1}{2}}}{1/2}\right) + C = ∫\sinh x dx$

$u^{\frac{1}{2}}+C=∫\sinh x dx$

Again substituting u = 1 + y², we get

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

And again Substitute y = sinhx here,

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

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

$\cosh x + C=∫\sinh x dx$

Hence the integral of sinhx is cosh x. Also, use the u-substitution calculator to solve the integral by using the substitution method.

Integral of sinhx 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 sinhx by using the indefinite integral.

Proof of integral of sinhx by using definite integral

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

$∫^π_0 \sinh x dx = \cosh x|^π_0$

Now, substituting the limit in the given function.

$∫^π_0 \sinh xdx=\cosh(π)-\cosh(0)$

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

$∫^π_0 \sinh x dx = -1 -1= -2$

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

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

Now,

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

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

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

Therefore, the definite integral of sinhx is equal to 1.