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186 LearnersLast updated on September 30, 2025
Integral of Sin 2x
Integration is one of the fundamental concepts of calculus. It is the inverse operation of differentiation. The symbol used to denote integral is โซ. In this topic, we will learn about the integrals of sin 2x, methods to find them, tips and tricks, and examples.
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.
Mistake 1
Incorrect use of substitution
Using the wrong substitution when integrating sin 2x, that is, u = sin 2x instead of u = 2x. The wrong substitution can lead to the wrong integration. So, students should understand that the inner function 2x should be substituted, which means let u = 2x. Then du = 2 dx or dx = du/2.
Mistake 2
Misinterpreting ∫sin 2x is sin x divided by 2
Students often think that ∫sin 2x is equal to ∫sin x divided by 2, which is wrong. So students should understand that the sin rule kx = -½ cos kx + C is not simply divided by 2. As k = 2, the ∫sin 2x = -½ cos 2x + C.
Mistake 3
Incorrect use of trigonometric identities
Students often apply and identify inappropriately or unnecessarily trigonometric identities.
For example, confusing with sin 2x = 2 sin x cos x. Even if the identity is correct, we won’t use it for the integration of sin 2x. Use the identity only if it simplifies the integral and verify the steps.
Mistake 4
Writing ∫sin 2x without dx
In the integration process, not adding the differential dx is a common error among students. Here, the dx indicates the variable with respect to the integration. So it is important to add dx to complete the integral expression, that ∫sin 2x dx
Mistake 5
Not verifying the answer
Most students won’t verify the answer after finding the value of integrals. To avoid errors, it is important to verify the answer by finding the differentiation of the antiderivative.
Examples of Integration of Sin 2x
Problem 1
Find the value of โซsin 2x dx
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
Problem 2
Evaluate the definite integral of sin 2x from 0 to ฯ/2
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
Problem 3
Find the value of โซsin(2x + 3)dx
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
Problem 4
Find the value of โซsinยฒ2x dx
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
Problem 5
Find the value of โซsin 2x cos 2x dx
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
FAQs on Integral of Sin 2x
1.What is the integral of sin 2 x?
2.What is the definite integral of sin 2x from 0 to ฯ/2
3.Is integral sin 2x dx the same as the integral of sinยฒdx?
4.What is C in integration?
5.What are the real-life applications of integrals?
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.
