Introduction to integral of sinh(ax)

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 sinh(ax) integral by using different integration techniques.

What is the integral of sinh(ax)?

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

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

Integral of sinh(ax) formula

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

$∫\sinh(ax)dx = \frac{\cosh(ax)}{a}+c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral.

How to calculate the integral of sinh(ax)?

$The integral of sinh(ax) 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(ax) 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 sinh(ax) by using derivatives.

Proof of integral of sinh(ax) by using derivatives

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

In derivative, we know that,

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

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

$∫\sinh(ax)dx=\frac{\cosh (ax)}{a}+c$

Hence the integral of sinh(ax) is equal to the cosh(ax)/a.

Integral of sinh(ax) by using substitution method

The u-substitution 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 sinh(ax) by using substitution method

To proof the integral of sinh(ax) by using substitution method, suppose that:

$y=\sinh(ax)$

Differentiating with respect to x,

$\frac{dy}{dx}=a\cosh(ax)$

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

$dy = a\cosh(ax)dx$

By trigonometric substitution identities, we know that cosh ax = √1 + sinh²(ax). Then the above equation becomes,

$dy = a\sqrt{1+ \sinh^2(ax)}.dx$

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

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

Multiplying both sides by sinh(ax),

$\frac{\sinh(ax) dy}{a\sqrt{1+y^2}}=\sinh(ax)dx$

Again substitute sinh(ax) = y on the left side.

$\frac{ydy}{a\sqrt{1+y^2}}=\sinh(ax)dx$

Integrating on both sides by applying integral,

$∫\frac{y dy}{a\sqrt{1 + y^2}}=∫\sinh(ax)dx$

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

Then the above left-hand side integral becomes,

$\frac{1}{2a}∫\frac{1}{\sqrt{u}}du=∫\sinh(ax)dx$

$\frac{1}{2a}∫u^{-\frac{1}{2}}du=∫\sinh(ax)dx$

Since the power rule of integration is

$∫x^ndx=\frac{x^{n+1}}{n+1}+C$

Therefore, by using this formula we get,

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

$\frac{u^{1/2}}{a}+C=∫\sinh(ax)dx$

Again substituting u = 1 + y², we get

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

And again Substitute y = sinh(ax) here,

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

$\frac{(\cosh^2ax)^{1/2}}{a}+C=∫\sinh(ax)dx$

$\frac{\cosh ax}{a}+C=∫\sinh(ax)dx$

Hence the integral of sinh(ax) is cosh x.

Integral of sinh(ax) 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 sinh(ax) by using the definite integral finder.

Proof of integral of sinh(ax) by using definite integral

To compute the integral of sinh(ax) 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 sinh(ax) from 0 to π. For this we can write the integral as:

$∫^π_0 \sinh(ax)dx=-\left|\frac{\cosh ax}{a}\right|^π_0$

Now, substituting the limit in the given function.

$∫^π_0 \sinh(ax)dx=\frac{\cosh (aπ)}{a}-\frac{\cosh a(0)}{a}$

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

$∫^π_0 \sinh(ax)dx=-\frac{1}{a}-\frac{1}{a}=-\frac{2}{a}$

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

$∫^{\frac{π}{2}}_0 \sinh(ax)dx=-\left|\frac{\cosh ax}{a}\right|^{\frac{π}{2}}_0$

Now,

$∫^{\frac{π}{2}}_0 \sinh(ax)dx=-\frac{\cosh a(π/2)}{a}+\frac{\cosh a(0)}{a}$

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

$∫^{\frac{π}{2}}_0 \sinh(ax)dx =0 -1/a=-1/a$

$Therefore, the definite integral of sinh(ax) is equal to -1/a.