Introduction to integral of sin(6x)

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area under a curve, 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(6x) integral by using different integration techniques.

What is the integral of sin 6x?

The integral of sin(6x) is an antiderivative of sine function which is equal to –cos 6x/6. 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 6x formula

The formula of integral of sin contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(sin(6x))dx. In mathematical form, the integration of sin 6x is:

$∫\sin(6x)dx =-\frac{\cos(6x)}{6}+c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. For any va;lue of n, we can calculate the sin nx integration.

How to calculate the integral of sin6x?

The sin6x integration is its antiderivative that can be calculated by using different integration techniques with integral calculator. In this article, we will discuss how to calculate integral of sine by using:

1. Derivatives
2. Substitution method
3. Definite integral

Integral of sin 6x 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(6x) by using derivatives.

Proof of integration of sin6x by using derivatives

Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integration of sin 6x by using its derivative. For this, we have to look for some derivatives formulas or a formula that gives sin(6x) as the derivative of any function.

In derivative, we know that,

$\frac{d}{dx}(\cos 6x)=-6\sin(6x)$

It means that the derivative of cos x gives us sin(6x). But it has negative sign. Therefore, to obtain the integral of sine, we have to multiply above equation by negative sign, that is:

$-\frac{d}{dx} (\cos 6x) = 6\sin(6x)$

Hence the integral of sin(6x) is equal to the negative of cos 6x. It is written as:

$∫\sin(6x)dx = -\frac{\cos(6x)}{6} + c$

Similarly, if we use the diffferentiation to derive the integral of sin 2t, it will be equal to -sin(2t)/2.

Integral of sin(6x) 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 sin6x integration by using substitution method

To prove the integral of sin6x by using u-substitution formula, suppose that:

$y=\sin(6x)\quad\text{and}\quad u= 6x$

then

$du = 6 dx$

The integral of sin (6x) will become,

$I=\frac{1}{6}∫\sin(u)du$

Now integrating with respect to u.

$I = -\frac{1}{6}\cos(u)+c$

Substituting the value of u back here, we get,

$I =-\frac{1}{6}\cos(6x)+c$

Which is the calculation of integral of sin(6x).

Integration of sin 6x 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(6x) by using the indefinite integral finder.

Proof of integral of sin(6x) by using definite integral

To compute the integration of sin6x by using a definite integral, we can use the interval from 0 to π or 0 to π/2. For example, the definite integral of sin(3x) is equal to 2/3. Let’s compute the integral of sin(6x) from 0 to π. For this we can write the integral as:

$∫^π_0 \sin(6x) dx = -\left|\frac{\cos 6x}{6}\right|^π_0 $

Now, substituting the limit in the given function.

$∫^π_0 \sin(6x)dx =-\frac{\cos 6(π)}{6} +\frac{\cos (0)}{6}$

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

$∫^π_0 \sin(6x) dx = -1 -1= -2$

Which is the calculation of the definite integral of sin(6x). Now to calculate the sin6x integration between the interval 0 to π/2, we just have to replace π by π/2. Therefore,

$∫^{\frac{π}{2}}_0 \sin(6x)dx =-\left|\frac{\cos 6x}{6}\right|^{\frac{π}{2}}_0$

Now,

$∫^{\frac{π}{2}}_0 \sin(6x)dx=-\frac{\cos 3π}{6} + \frac{\cos (0)}{6}$

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

$∫^{\frac{π}{2}}_0 \sin(6x) dx = 0 + 1=1$

Therefore, the definite integral of sin(6x) is equal to 1.