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Integral of Sin X: Definition, Formula, Proof, and Examples

The integral of sin x means reversing differentiation to get -cos(x) + C. Check out the integral of sin x formula, proof, and examples for a better understanding of this calculus concept.
authorImageChandni 6 Aug, 2025
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Integral of Sin x

Before finding the Integral of Sin⁡ X , let’s first understand what integration is. Integration is the reverse process of differentiation. If differentiation helps us find how a function changes, integration helps us find the original function from its rate of change. Simply put:

  • Differentiation gives the slope of a function.
  • Integration helps us work backward to find the function itself.
In this blog, we will discuss the concept of the integral of sin⁡(x) in detail. 

What is the Integral of Sin⁡ X?

Lets understand What is the integration of sin x dx. The integral of sin⁡(x) is −cos⁡(x), and mathematically, it is expressed as: ∫sin⁡(x) dx = −cos⁡(x) + C Where C is the constant of integration. Here's what the components mean:
  • ∫: The symbol for integration.
  • Sin⁡(x): The function being integrated, known as the integrand.
  • dx: Represents a small change in x, indicating the variable of integration.
To solve ∫sin⁡(x) dx, there are different approaches, such as: Using derivatives: Reversing the process of differentiation. Substitution method: Transforming the integral into an easier form. We'll explore these methods in detail in the following sections. 

Integral of Sin⁡ X Formula

The sin x integration formula is: ∫sin⁡(x) dx = −cos⁡(x) + C Here’s what it means:
  • Integration is like the reverse of differentiation. When you differentiate −cos(x), you get sin(x). That’s why −cos(x) is the integral.
  • C is the constant of integration. It represents any constant that could have been in the original function because the derivative of any constant is zero.

Integral of Sin X Graphical Representation

The integral of sin(x) has clear graphical significance, as it represents the area under the curve of sin(x) between two points a and b. This area can be calculated using the definite integral method and the geometrical method .

Definite Integral Method

Using calculus, the integration of sinx from a to b is expressed as:
 This sin integration formula gives the signed area between the curve and the x-axis:
  • Positive area is above the x-axis (sin⁡(x) > 0).
  • Negative area is below the x-axis (sin⁡(x) < 0)

Example:

To find the area from a = 0 to b = π/2b ∫ 0 π/2 sin x dx = [-cos x]₀ π/2 = -cos π/2 - (-cos 0) = -(0) + 1 = 1 Here, the area under the sine curve from 0 to π/2 is exactly 1.

Geometrical Method

To visualize the area under the sine curve, consider the graph of sin(x) between a and b. The area can be divided into positive regions (above the x-axis) and negative regions (below the x-axis). The total area is the sum of these regions.

Example:

From 0 to π/2, the curve of sin(x) forms a smooth quarter circle. The geometrical area is approximately calculated as: Area = ¼ × Area of Circle with Radius 1. However, the exact calculation using integration confirms the area is 1 . For the interval 0 to π, where sin⁡(x) rises to 1 at π/2 and falls back to 0 at π, the total area is: ∫₀ π sin x dx = [-cos x]₀ π = -cos π - (-cos 0)] As cos(π) is -1 and cos(0) is 1, the expression simplifies to: ∫ 0 π sin(x) dx = [-(-1) + 1] Thus, the integral of sin x from 0 to π is 2.

Integration of Sin X Proof Using the Substitution Method

To calculate ∫sin(x) dx  using the substitution method, follow these simple steps: Choose a Substitution : Let u = cos⁡(x) Differentiate u with respect to x: du/dx = −sin⁡(x) This gives: du = −sin⁡(x) dx. Solve for dx: dx = du/ −sin⁡(x) Since du = -sin(x) dx , we can substitute this into the original integral. ∫sin⁡(x) dx

Rewriting using substitution for integration of sinx:

∫sin⁡(x) dx = −∫du. Integrate with respect to u: −∫du = −u + C. Substituting back u = cos⁡(x) Since we set u = cos⁡(x) in Step 1, we substitute it back: − cos ( x ) + C Thus, the integral of sin⁡(x) is −cos⁡(x)+C

Integration of Sin X Solved Examples

Example 1: Find the integral of sin⁡(3x) Solution: We use the substitution method for this integral: ∫sin⁡(3x) dx Let u = 3xu, so du = 3 dx or dx = du/3 Substitute u and dxn into the integral: ∫sin⁡(3x) dx = ∫sin⁡(u) × 1/3 du = 1/3∫sin⁡(u) du. The integral of sin⁡(u) is −cos⁡(u) 1/3∫sin⁡(u) du = 1/3(−cos⁡(u))+C. Replace u = 3xu back:  ∫sin⁡(3x) dx=−1/3/cos⁡(3x)+C.  
Example 2: Find the integral of sin⁡  2 (2x) Solution: To integrate sin⁡ 2 ( 2x), use the trigonometric identity: sin⁡ 2 (2x) = 1/2(1−cos⁡(4x) Substitute this identity into the integral: ∫sin⁡ 2 (2x) dx = ∫1/2(1−cos⁡(4x)) dx. Separate the terms: ∫sin⁡ 2 (2x) dx = 1/2∫1 dx−1/2∫cos⁡(4x) dx. Integrate each term: The integral of 1 is x. The integral of cos⁡(4x) is sin⁡(4x)/4. So: ∫sin⁡ 2 (2x) dx = 1/2x−1/2×sin⁡(4x)/4+C. Simplify: ∫sin⁡ 2 (2x) dx = x /2 −sin⁡(4x)/8+C.
Example 3: Find the integral of sin⁡ 3 (5x) Solution: To integrate sin⁡ 3 (5x), use the identity sin⁡ 3 (5x) = sin⁡(5x)⋅sin⁡ 2 , and rewrite sin⁡ 2 (5x) as 1−cos⁡ 2 (5x): ∫sin⁡ 3 (5x) dx =∫sin⁡(5x)×(1−cos⁡ 2 (5x)) Let u = cos⁡(5x), so du =−5sin⁡(5x) dx or dx = -du/- 5sin(5x) Substitute u into the integral: ∫sin⁡ 3 (5x) dx =−1/5∫(1−u 2 ) du. Expand and integrate: −1/5∫(1−u 2 ) du =−1/5(u−u 3 /3) + C Replace u = cos⁡(5x) back: ∫sin⁡ 3 (5x) dx = −1/5 (cos⁡(5x) − cos⁡ 3 (5x)/3) + C. Join Kids Online Tuition Class Now!!

Integral of Sinx FAQs

How does ∫sin⁡(x) dx differ from differentiation?

The integral ∫sin⁡(x) dx and differentiation of sin⁡(x) are opposite processes. Differentiation finds the rate of change or slope of a function, while integration reverses this process to retrieve the original function (up to a constant). For example: Differentiating −cos(x) gives sin(x) Integrating sin⁡(x) gives −cos⁡(x)+C, where C accounts for any constant lost during differentiation

What is the physical significance of integrating sin(x) in engineering?

Integration of sin(x) is often used to analyze waveforms, oscillatory motions, and signal processing.

What is the geometric meaning of ∫sin(x)dx?

It represents the area under the sine curve, considering both positive and negative regions.

What is the relationship between sin(x) and cos(x) in integration?

The integral of sin⁡(x) is −cos⁡(x), while the derivative of cos⁡(x) is −sin⁡(x).
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