Introduction to integral of sin(3x)

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 sin 3x. You will also understand how to compute sin3x integral by using different integration techniques.

What is the integral of sin3x?

The integral of sin(3x) is an antiderivative of sine function which is equal to –cos x. 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 3x formula

The formula of integral of sin contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫sin(3x)dx. In mathematical form, the integration of sin3x is:

$∫\sin(3x)dx = -\frac{\cos 3x}{3} + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. Moreover, by for any value of n, you can calculate the integral of sin nx by using above formula.

How to evaluate integral of sin(3x) with respect to x?

The antiderivative of sin 3x 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 sin3x 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(3x) by using derivatives.

Proof of integral of sin3x by using derivatives

Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integral of sin 3x by using its derivative. For this, we have to look for some derivatives formulas or a formula that gives sin 3x as the derivative of any function.

In derivative, we know that,

$\frac{d}{dx}(\cos 3x) = -3\sin(3x)$

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

$-\frac{d}{dx}(\cos 3x) = 3\sin(3x)$

Hence the integral of sin(3x) is equal to the negative of 3cos 3x. It is written as:

$∫\sin(3x)dx = -\frac{\cos (3x)}{3} + c$

Antiderivative of sin 3x 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 sin3x by using the substitution method.

Proof of integral sin3x by using substitution method

In the substitution method, we can use trigonometric identities as well as parameters so that we can write the equation in simple form. This method helps to calculate integrals easily. To proof the integration of sin3x by using substitution method, suppose that:

$y = \sin (3x)$

Using integral, suppose that u = 3x and du = 3dx, therefore, the integral of sin (3x) can be written as:

$∫\sin (3x)dx = \frac{1}{3}∫\sin u du$

Integrating with respect to the variable involved we get,

$∫\sin (3x) dx =-\frac{\cos u}{3} + c$

Now substituting u = 3x to get a solution.

$∫\sin (3x)dx=-\frac{\cos 3x}{3} + c$

Hence we have verified the integral of sin (3x) by using a substitution method. You can also use the u-substitution online calculator to integrate sin 3x.

Integration of sin 3x 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 sin(3x) by using the indefinite integral.

Proof of integration of sin3x by using definite integral

To compute the antiderivative of sin 3x 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(3x) from 0 to π. For this we can write the integral as:

$∫^π_0 \sin(3x) dx =-\left|\frac{\cos 3x}{3}\right|^π_0$

Now, substituting the limit in the given function.

$∫^π_0 \sin(3x) dx = -\frac{\cos 3(π)}{3} +\frac{\cos 3(0)}{3}$

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

$∫^π_0 \sin(3x) dx = \frac{1}{3}+\frac{1}{3}= \frac{2}{3}$

Which is the calculation of the definite integral sin3x. Now to calculate the integral of sin 3x between the interval 0 to π/2, we just have to replace π by π/2. Therefore,

$∫^{\frac{π}{2}}_0 \sin(3x)dx = -\left|\frac{\cos 3x}{3}\right|^{\frac{π}{2}}_0$

Now,

$∫^{\frac{π}{2}}_0 \sin(3x)dx = -\frac{\cos 3(π/2)}{3} +\frac{\cos 3(0)}{3}$

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

$∫^{\frac{π}{2}}_0 \sin(3x) dx = 0 + \frac{1}{3} = \frac{1}{3}$

Therefore, the definite integral of sin(3x) is equal to 1/3.