Integral of Sinx

Integral of sinx along with its formula and proof with examples. Also learn how to calculate integration of sinx with step by step examples.

Alan Walker-

Published on 2023-04-13

Introduction to the integral of sin x

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 calculator 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 formula sinx by using different integration techniques.

What is the integral of sin?

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

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

Sin = opposite side/hypotenuse

Integration of sinx 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)dx. In mathematical form, the sinx integration formula is:

$∫(\sin x)dx = -\cos x + c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. It may be noted that the integral of cos x is equal to sin x. 

How to calculate the integral of sin(x)?

The integral of sin x 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 sinx 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 by using derivatives.

Proof of integral of sin x by using derivatives

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

In derivative, we know that,

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

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

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

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

$∫(\sin x)dx = -\cos x + c$

Integration of sin x by using u substitution method

The substitution method calculator online 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 sinx integration by using substitution method

To prove the integral of sin x by using the substitution method, suppose that:

$y = \sin x$

Differentiating with respect to x,

$\frac{dy}{dx} = \cos x$

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

$dy = \cos x dx$

By trigonometric identities, we know that cos x = √1 - sin²x. Then the above equation becomes,

$dy = \sqrt{1 - \sin²x}.dx$

Now, substituting the value of sin2 x, such as:

$dy = \sqrt{1 – y^2}.dx$

Multiplying both sides by sin x,

$\frac{\sin x dy}{\sqrt{1 - y²}} = \sin x dx$

Again substitute sin x = y on the left side.

$\frac{ydy}{\sqrt{1 - y²}} = \sin x dx$

Integrating on both sides by applying integral,

$∫\frac{ydy}{\sqrt{1 - y²}} = ∫\sin x dx$

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

Then the above left-hand side integral becomes,

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

$(-1/2)∫u^{-1/2}du=∫\sin xdx$

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

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

$-u^{1/2} + C = ∫\sin x dx$

Again substituting u = 1 - y², we get

$-(1 - y²)^{1/2} + C = ∫\sin x dx$

And again Substitute y = sin x here,

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

$-(\cos²x)^{1/2} + C = ∫\sin x dx$

$-\cos x + C=∫\sin x dx$

Hence the sinx integration is –cos x. Also, the trigonometric substitution calculator is helpful to calculate integral of non-linear functions by using trigonometric functions.

Sin x 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:

$∫^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integral of sinx by using the definite integral.

Proof of antiderivative of sinx by using definite integral

To compute the integration of sin x 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 sine from 0 to π. Since the indefinite integral of sin x is cos x, this we can write the integral as:

$∫^π_0 \sin x dx = -\cos x|^π_0$

Now, substituting the limit in the given function.

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

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

$∫^π_0 \sin x dx = -1 -1= -2$

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

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


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

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

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

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

Frequently Asked Questions

Can you integrate sin2x?

Integration of sin2x means finding the integral of the function sin2x. Integral of sin2x can be written as ∫ sin2x dx. Here, we need to find the indefinite integral of sin2x. So, the integration of sin2x results in a new function with arbitrary constant C.

What is integral 2x?

The integration of 2x in calculus is equal to x square plus the constant of integration which is symbolically written as ∫2x dx = x2 + C, where ∫ is the symbol of the integral, dx shows that the integration of 2x is with respect to the variable x and C is the constant of integration.

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