Integral of Sin 2x
2026-02-28 01:11 Diff
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Last updated on September 30, 2025

Integration is one of the fundamental concepts of calculus. It is the inverse operation of differentiation. The symbol used to denote integral is ∫. In this topic, we will learn about the integrals of sin 2x, methods to find them, tips and tricks, and examples.

What is the Integral of Sin 2x

The integral of sin 2x is -(cos 2x) / 2 + C, and C here is the constant of integration. It can be written as ∫sin 2x dx. There are different methods to solve the integral of sin 2x. Let’s learn the different methods to solve integration.  

Methods to Solve the Integral of Sin 2x

To find the value of ∫sin 2x, we use the substitution method. The substitution method is the technique used to simplify the integral by changing the independent variable.

Now, let’s see how to find the value of ∫sin 2x using the substitution method.

Let’s consider 2x = u

Then 2 dx = du, that is dx = du/2

Substituting the value in ∫sin 2x,

∫sin 2x dx = ∫ sin u(du/2)

=½ ∫sin u du

Since the integral of sin x is -cos x + C, we get,

½ ∫sin u du = ½ (-cos u) + C

As u = 2x, substituting the value,

∫sin 2x dx = -(cos 2x) / 2 + C

Tips and Tricks for Integration of Sin 2x

Integration is considered a hard topic in math by students. Let’s learn a few tips and tricks to master the integration of sin 2x. 

  • To find the value of ∫sin 2x dx, let’s break it down with substitution. So let’s consider 
    u = 2x, so du = 2dx then dx = du/2
    ∫sin 2x dx = ∫sin u du/2
    = -½ cos u + C
    As u = 2x,
    Then, -½ cos u + C is -½ cos 2x + C
  • The integral of sine always gives a cosine term and is divided by the coefficient of x in the sine function. Add a negative sign, ∫sin 2x dx = -½ cos 2x + C
  • Practice and verify the integration answer by differentiating. d/dx (-½ cos 2x) is sin 2x. 
  • Use the mnemonics such as “sine integrates to negative cosine, don’t forget to divide the line!”

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Common Mistakes and How to Avoid Them in the Integration of Sin 2x

Students often consider mathematics as a difficult topic, that too integration. So, to master integration, let's look at a few common mistakes and ways to avoid them.

Problem 1

Find the value of ∫sin 2x dx

Okay, lets begin

The value of ∫sin 2x dx is -½ cos 2x + C

Explanation

Using the substitution method, 

Let u = 2x

Then, du = 2dx

So, dx = du/2 

∫sin 2x dx = ∫sin u du/2

= ½ ∫sin u du

As  ∫sin u du = -cos u + C

½ (-cos u) + c = -½ cos u + c

As u = 2x,

-½ cos 2x + c

Well explained 👍

Problem 2

Evaluate the definite integral of sin 2x from 0 to π/2

Okay, lets begin

Integral of sin 2x from 0 to π/2is 1

Explanation

The antiderivative of ∫sin 2x dx is -½ cos 2x + C

Evaluate x = 0 to x = π/2 

[-½ cos 2x]0π/2= [-½ cos π] - [-½ cos 0]

Since cos π = -1 and cos 0 = 1

-½ (-1) - [-½ (1)] = ½ + ½  = 1

Well explained 👍

Problem 3

Find the value of ∫sin(2x + 3)dx

Okay, lets begin

The value of ∫sin(2x + 3)dx = -½ cos(2x + 3) + C

Explanation

Let u = 2x + 3

Then du = 2dx; then dx = du/2 

∫sin(2x + 3)dx = ∫sin u du/2 

= ½ ∫sin u du

With ∫sin u du = -cos u + C

½ (-cos u) + C = -½ cos u + C

Substitute back u = 2x + 3

= -½ cos (2x + 3) + C

Well explained 👍

Problem 4

Find the value of ∫sin²2x dx

Okay, lets begin

The value of ∫sin22x dx is (x/2) - (sin 4x/8) + C

Explanation

Using the trigonometric identity

Sin22x = 1 - cos 4x / 2

∫sin22x dx = ∫1 - cos 4x / 2 dx 

= ½ ∫cos 4x dx

Integrate term by term

½ ∫dx = x/2

∫cos 4x dx = sin 4x / 4

So, ∫sin22x dx = x/2 - ½(sin 4x/4) + C

= x/2 - sin 4x/8 + C

Well explained 👍

Problem 5

Find the value of ∫sin 2x cos 2x dx

Okay, lets begin

The value of ∫sin 2x cos 2x dx = - cos 4x / 8 + C

Explanation

The double-angle identity for sine:

Sin 2x cos 2x = ½ sin4x

Then, ∫sin 2x cos 2x dx = ∫1/2 sin 4x dx

= ½ ∫sin 4x dx

Now, integrate: ∫sin 4x dx = -cos 4x / 4 + C

So, ½ (-cos 4x/ 4) +C 

= -cos 4x / 8 +C

Well explained 👍

FAQs on Integral of Sin 2x

1.What is the integral of sin 2 x?

The integral of sin 2x is -½ cos 2x + c

2.What is the definite integral of sin 2x from 0 to π/2

We write the integral of sin 2x from 0 to π/2 as ∫ 0π/2sin 2x dx. The value of ∫ 0π/2sin 2x dx is 1

3.Is integral sin 2x dx the same as the integral of sin²dx?

No, ∫sin 2x dx and ∫sin2dx are not the same. 

4.What is C in integration?

The C in integration represents the constant of integration.

5.What are the real-life applications of integrals?

Integration is used in various fields like physics, engineering, economics, biology, medicine, probability, and statistics.

Important Glossaries for Integration of Sin 2x

  • Antiderivative: The result of the integration of a function is the antiderivative. Whose derivative gives the original function. For example, ∫ sin 2x = -½ cos 2x + c, here -½ cos 2x + c is the antiderivative.
  • Substitution method: A method used to find the value of integration; here, a substitution is used to simplify the integrals.  
  • Constant of integration (C): The constant of integration is a constant-used integration. It represents a number that could be added to the integral function.
  • Trigonometric identity: The formula is used to simplify the trigonometric expression.

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