Integral Of Sin(Nx)

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

Alan Walker-

Published on 2023-04-13

Introduction to integral of sin(nx)

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 integration of sinnx by using different integration techniques.

What is the integration of sin nx?

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

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

$∫\sin nxdx=-\frac{\cos nx}{n}+c$

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

How to calculate the integral of sinnx?

The sin nx integration 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 sin(nx) 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. The derivatives can also be used to solve many integral problems. Let’s discuss calculating the sinnx integration by using derivatives.

Proof of integral of sin(nx) formula by using derivatives

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

In derivative, we know that,

$\frac{d}{dx} (\cos nx) = -n\sin nx$

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

$-\frac{d}{dx}(\cos nx) = n\sin nx$

Hence the integral of sin nx is equal to the negative of cos nx. It is written as:

$∫\sin nxdx=-\frac{\cos nx}{n}+c$

Sinnx integration 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 formula.

Proof of integration of sin nx by using substitution method

To proof the integral of sin(nx) formula by using substitution method, suppose that:

$y = \sin nx$

In integral form,

$I=∫\sin nxdx$

Suppose that nx = t and ndx = dt. Now substituting these in above equation.

$I = ∫\frac{\sin t}{n}dt$


$I=\frac{1}{n}∫\sin tdt$

Now integrating with respect to t, we get,

$I=-\frac{1}{n}\cos t + c$

Now by substituting the value t back here, we can get the integral of sin nx by trig substitution method. Hence,

$I=-\frac{1}{n}\cos nx + c$

Which is the calculation of integral of sinnx.

Sin nx integration 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){2}lt;/p>

Let’s understand the verification of the sinnx integration by using the indefinite integral.

Proof of integration of sinnx by using definite integral

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

$∫^π_0 \sin nx dx = -\frac{\cos nx}{n}|^π_0$

Now, substituting the limit in the given function.

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

Here, we consider two cases where n is even and odd. Suppose n is odd then the value of cos nπ is -1 therefore,

$∫^π_0 \sin nx dx = -1 +1=0$

Now when n is an even number then the value of cos nπ will be 1. Therefore,

$∫^π_0 \sin nx dx = 1 +1= 2$

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

$∫^{\frac{π}{2}}_0 \sin nx dx=-\left|\frac{\cos nx}{n}\right|^{\frac{π}{2}}_0$


$∫^{\frac{π}{2}}_0 \sin nxdx=-\frac{\cos n(π/2)}{n} + \frac{\cos n(0)}{n}$

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

$∫^{|frac{π}{2}}_0 \sin nx dx = 0$

Therefore, the definite integral of sinnx is equal to 0. You can use the definite integration solver to compute the integration of sin nx.

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