Integral Of Sin(Ax)

Integral of sin(ax) along with its formula and proof with examples. Also learn how to calculate integration of sin ax with step by step examples.

Alan Walker-

Published on 2023-04-13

Introduction to integral of sin(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 sin(ax) integral by using different integration techniques.

What is the integration of sinax?

The integral of sinax is an antiderivative of sine function which is equal to –cos(ax)/a. It is also known as the reverse derivative of sine function which is a trigonometric identity.

The sine function is the ratio of opposite side to the hypotenuse of a triangle which is written as:

Sin = opposite side / hypotenuse

Integral of sin(ax) formula

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

$∫\sin(ax)dx = -\frac{\cos ax}{a}+c$

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

How to calculate the sinax integration?

The integration of sin 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 sinax 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 sin(ax) by using derivatives.

Proof of integration of sin ax by using derivatives

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

In derivative, we know that,

$\frac{d}{dx}(\cos ax) = -a\sin(ax)$

It means that the derivative of cos 3a gives us sin 3a. But it has negative sign. Therefore, to obtain the integral of sine, we have to multiply above equation by negative sign, that is:

$-\frac{d}{dx}(\cos ax) = a\sin(ax)$

Hence the integral of sin ax is equal to the negative of 3cos 3a. It is written as:

$∫\sin(ax)dx = -\frac{\cos (ax)}{a}+c$

Integral of sin(ax) 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 sin ax integration by using substitution method

In substitution method, we can use trigonometric identities as well as parameters so that we can write the equation is simple form. This method helps to calculate integral easily. To proof the sinax integration by using substitution method, suppose that:

$y = \sin(ax)$

Using integral, suppose that u = ax and du = adx, therefore, the integral of sin (ax) can be written as:

$∫\sin(ax)dx = \frac{1}{a}∫\sin u du$

Integrating with respect to the variable involved we get,

$∫\sin (ax) dx =-\frac{1}{a}.\cos u + c$

Now substituting u = ax to get solution.

$∫\sin (ax) dx=-\frac{1}{a}\cos(ax) + c$

Hence we have verified the integral of sin (ax) by using substitution method.

Integral sin 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 sinax by using the definite integral calculator.

Proof of integral of sin ax by using definite integral

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

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

Now, substituting the limit in the given function.

$∫^π_0 \sin(ax)dx = -\frac{\cos a(π)}{a} + \frac{\cos a(0)}{a}$

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

$∫^π_0 \sin(ax) dx = 1 +1= 2$

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

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


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

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

$∫^{\frac{π}{2}}_0 \sin(ax) dx = 0 + 1 = 1$

Therefore, the definite integration of sinax is equal to 1.

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