Integration of sin 2x and sin²x: Formulas, Derivations & Solved Examples

Integration can be referred to as the inverse operation of differentiation. We know that by differentiating we divide a big part into a number of smaller parts. In integration, we combine smaller parts to form a larger whole. This can also be called the antiderivative.

Generally, the process of integration is used for finding areas. When this area is found under a specified limit or boundary , it is termed as a definite integral. In the definite integral the upper and the lower limits of independent variables in a function are specified. 

Mathematically, any definite integral can be represented as:

However, in the case of indefinite integral, the limit or the boundary is not specified and a constant C is added after finding the value of integration. 

In this maths article, we shall learn about integration of sin 2x and sin^2x in different limits. We shall also check different methods to derive the formula for integration and some solved examples for better understanding of the concept. 

What is the integral of sin 2x dx?

We denote the integral of sin 2x as and its value is , here C is the constant of integration. 

To prove this, we follow substitution method of integration:

Let us assume 2x = u. Then 2 dx = du, or, dx = du/2.

Now, substituting the values in the integral . 

=

Also we know that the integral of sin x = -cos x + C

So, 

= (1/2)(-cos u) + C

Substituting u = 2x, so

Therefore, this is the formula for integral of sin 2x.

An indefinite integral with some lower and upper bounds are termed as definite integrals. According to the fundamental theorem of Calculus, the value of the upper bound and the lower bound is substituted in the value of indefinite integral and then is subtracted in the same order. 

When we find the value of a definite integral, the integration constant can be ignored. Let us check the values of some definite integral here:

Integral of sin 2x from 0 to

=

=

=1

Therefore, the value of the integral of sin 2x in the limit 0 to is 1.

Integral of sin 2x from 0 to

=0

Therefore, the value of the integral of sin 2x in the limit 0 to is 0.

What is the integral of dx?

We denote the integral of as . The value of integral of is given as .

This value can be found using two methods:

• Using the formula for cos 2x
• Using the formula for integration by parts.

Method 1: Integral of Using Double Angle Formula of Cos x

We can use the double angle formula for cos x to find the value of . We know that one of the double angle formula for cos x is given as:

When we solve this for , we get:

Now, we can find the value of

We get:

Also, we know that:

Therefore,

This is the integral formula for .

Method 2: Integral of Using Integration by Parts

We know that can be written as sin x . sin x. So, to find the integral of the product, we can use the formula for integration by parts.

Here, u = sin x and dv = sin x dx

Then du = cos x dx, and v = -cos x

Using the formula for integration by parts:

Also, we can use the double angle formula for sin 2x;

We know that 2 sin x cos x = sin 2x

Also, using trigonometric identity, we know that

, here,

Hence Proved

Definite Integral of

In order to find the value of the definite integral of we simply substitute the value of the lower and upper bound of the limits in the value of the integral of and then subtract the values obtained. 

Integral of from 0 to

=

Therefore, the integral of in the limits 0 to is

Integral of from 0 to

Therefore, the integral of in the limits 0 to is

Summary

• The value of is , here C is the constant of integration. 
• The value of the integral of sin 2x in the limit 0 to is 1.
• The value of the integral of sin 2x in the limit 0 to is 0.
• The value of the integral of is given as .
• The integral of in the limits 0 to is
•  The integral of in the limits 0 to is

Properties of Integration of sin 2x:

1. Basic Formula:
   ∫ sin(2x) dx = –½ cos(2x) + C
   (This is the standard result when integrating sin(2x).)
2. Linearity Property:
   If a is a constant, then:
   ∫ a · sin(2x) dx = a · ∫ sin(2x) dx = –a/2 · cos(2x) + C
3. Sum and Difference Rule:
   ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
   This allows you to integrate sin(2x) as part of a larger expression.
4. Substitution Method:
   To solve ∫ sin(2x) dx, use substitution:
   Let u = 2x → du = 2 dx → dx = du/2
   Then, ∫ sin(u) · (du/2) = –½ cos(u) + C = –½ cos(2x) + C
    
5. Even Function Property:
   sin(2x) is an odd function, so:
   ∫ from –a to a of sin(2x) dx = 0
   (Useful for definite integrals over symmetric intervals.)

Integration of sin 2x dx Solved Examples

Q 1: Find the value of the integral

Ans 1: Given that:

We need to find the value of

We can write the given integral as:

Also, we know that:

So, the above integral becomes:

Let us assume sin x = u

Then cos x dx = u du

So,

Now, substituting u = sin x

Therefore, the integral of is .

Q 2: Find the value of the integral

Ans 2: Give that:

We need to find the value of the integral

Let us solve this question using substitution method:

Let cos 2x = u, therefore, -2sin 2x dx = du

Or, sin 2x dx = (-1/2)du

So, the given integral becomes:

Now, let us replace the value of u as cos 2x here;

So, =

Therefore, the integral of is .

Q 3: Find the value of the integral: ∫ (sin 4x) / (1 + cos 2x) dx

Solution:
We are given the integral:
∫ (sin 4x) / (1 + cos 2x) dx

We know that:
sin 4x = 2 sin 2x cos 2x
So the integral becomes:
∫ [2 sin 2x cos 2x] / (1 + cos 2x) dx

= 2 ∫ [cos 2x / (1 + cos 2x)] × sin 2x dx

Now, let’s make a substitution:
Let u = cos 2x
Then, du = -2 sin 2x dx
So, sin 2x dx = -½ du

Substitute back into the integral:
= 2 ∫ [u / (1 + u)] × (–½ du)
= –∫ [u / (1 + u)] du

Now, rewrite the fraction:
u / (1 + u) = (u + 1 – 1) / (1 + u) = 1 – 1 / (1 + u)

Now integrate:
∫ (1 – 1 / (1 + u)) du = ∫ 1 du – ∫ 1 / (1 + u) du
= u – ln|1 + u| + C

Now substitute u = cos 2x:
= cos 2x – ln|1 + cos 2x| + C

Final Answer:
∫ (sin 4x) / (1 + cos 2x) dx = cos 2x – ln|1 + cos 2x| + C

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