Introduction to the the integral of sin x by 2

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area under the 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(x/2) integral by using different integration techniques.

What is the integral of sin (x/2)?

The integral of sin(x/2) is an antiderivative of sine function which is equal to –2cos x/2. It is also known as the reverse derivative of sine function, 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 x/2 formula

The formula of the integral of sin contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(sin(x/2))dx. In mathematical form, the integral of sin(x/2) is:

$∫\sin(x/2)dx = -2\cos(x/2) + c$

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

How to calculate the integral of sin(x/2)?

The integral of sin(x/2) is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the integral of sine by using:

1. Derivatives
2. Substitution method
3. Definite integral

Integral of sin(x/2) 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(x/2) by using derivatives.

Proof of integral of sin(x/2) by using derivatives

Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integral of sin(x/2) by using its derivative. For this, we have to look for some derivative formulas or a formula that gives sin(x/2) as the derivative of any function.

In derivative, we know that,

$\frac{d}{dx} (\cos x/2) = -\frac{\sin(x/2)}{2}$

It means that the derivative of cos x gives us sin(x/2). But it has a negative sign. Therefore, to obtain the integral of sine, we have to multiply the above equation by the negative sign, that is:

$-\frac{d}{dx} (\cos x/2) = \frac{\sin(x/2)}{2}$

Hence the integral of sin(x/2) is equal to the negative of 2cos x/2. It is written as:

$∫\sin(x/2)dx = -2\cos(x/2) + c$

Integral of sin(x/2) 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 calculator.

Proof of Integral of sin(x/2) by using substitution method

To prove the integral of sin(x/2) by using the substitution method, suppose that:

$y = \sin(x/2)$

Differentiating with respect to x,

$\frac{dy}{dx} = \frac{\cos(x/2)}{2}$

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

$dy = \frac{\cos(x/2)}{2} dx$

By trigonometric identities, we know that cos(x/2) = √1 - sin²(x/2). Then the above equation becomes,

$2dy = \sqrt{1 - \sin²(x/2)}.dx$

Now, substituting the value of sin2(x/2), such as:

$2dy = \sqrt{1 – y^2}dx$

Multiplying both sides by sin(x/2),

$\frac{2\sin(x/2) dy}{\sqrt{1 - y²}} = \sin(x/2) dx$

Again substitute sin(x/2) = y on the left side.

$\frac{2y dy}{\sqrt{1 - y²}} = \sin(x/2) dx$

Integrating on both sides by applying integral,

$∫ \frac{2ydy}{\sqrt{1 - y²}} = ∫\sin(x/2) dx$

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

Then the above left-hand side integral becomes,

$-∫ \frac{1}{√u}du = ∫ \sin(x/2) dx$

$-∫ u^{-1/2}du = ∫\sin(x/2) dx$

Since the power rule of integration is ∫ x^n dx = (x^n+1)/(n+1) + C. Therefore, by using this formula we get,

$-2(u^{1/2}) + C = ∫ \sin(x/2) dx$

$-2u^{1/2} + C = ∫ \sin(x/2) dx$

Again substituting u = 1 - y², we get

$-2(1 - y²)^{1/2} + C = ∫ \sin(x/2) dx$

And again Substitute y = sin(x/2) here,

$-2(1 - \sin²x/2)^{1/2} + C = ∫ \sin(x/2) dx$

$-2(\cos²x/2)^{1/2} + C = ∫ \sin(x/2) dx$

$-2\cos(x/2) + C = ∫ \sin(x/2) dx$

Hence the integral of sin(x/2) is –2cos x/2. Finding the integral by using the trigonometric substitution formula can be easier. 

Integral of sin(x/2) 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(x/2) by using the indefinite integral.

Proof of integral of sin(x/2) by using definite integral

To compute the integral of sin(x/2) 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(x/2) from 0 to π. For this we can write the integral as:

$∫^π_0 \sin(x/2) dx = -\cos(x/2)|^π_0$

Now, substitute the limit in the given function.

$∫^π_0 \sin(x/2) dx = -\cos (π/2) + \cos (0)$

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

$∫^π_0 \sin(x/2) dx = 0+1= 1$

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

$∫^{π/2}_0 \sin(x/2) dx = -2\cos(x/2)|^{π/2}_0$

Now,

$∫^{π/2}_0 \sin(x/2) dx = -2\cos (π/4) + 2\cos (0)$

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

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

Therefore, the definite integral of sin(x/2) is equal to √2. Try our definite integration calculator. It is free to use.