# Just a moment

The integral of sin 2x & the integral of sin2x have different values. Lớn find the integral of sin2x, we use the cos 2x formula & the substitution method whereas we use just the substitution method to find the integral of sin 2x.

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Let us identify the difference between the integral of sin 2x and the integral of sin2x by finding their values using appropriate methods & also we will solve some problems related khổng lồ these integrals.

 1 What is the Integral of sin 2x dx? 2 Definite Integral of sin 2x 3 What is the Integral of sin^2x dx? 4 Definite Integral of sin^2x 5 FAQs on Integral of sin 2x & sin^2x

## What is the Integral of Sin 2x dx?

The integral of sin 2xis denoted by ∫ sin 2x dx and its value is -(cos 2x) / 2 + C, where 'C' is the integration constant. For proving this, we use the integration by substitution method. For this, we assume that 2x = u. Then 2 dx = du (or) dx = du/2. Substituting these values in the integral ∫ sin 2x dx,

∫ sin 2x dx = ∫ sin u (du/2)

= (1/2) ∫ sin u du

We know that the integral of sin x is -cos x + C. So,

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

Substituting u = 2x back here,

∫ sin 2x dx = -(cos 2x) / 2 + C

This is the integral of sin 2x formula.

## Definite Integral of Sin 2x

A definite integral is an indefinite integral with some lower and upper bounds. By the fundamental theorem of Calculus, khổng lồ evaluate a definite integral, we substitute the upper bound và the lower bound in the value of the indefinite integral và then subtract them in the same order. While evaluating a definite integral, we can ignore the integration constant. Let us calculate some definite integrals of integral sin 2x dx here.

### Integral of Sin 2x From 0 lớn pi/2

∫(_0^pi/2) sin 2x dx = (-1/2) cos (2x) (left. ight|_0^pi/2)

= (-1/2)

= (-1/2) (-1 - 1)

= (-1/2) (-2)

= 1

Therefore, the integral of sin 2x from 0 lớn pi/2 is 1.

### Integral of Sin 2x From 0 to lớn pi

∫(_0^pi) sin 2x dx = (-1/2) cos (2x) (left. ight|_0^pi)

= (-1/2)

= (-1/2) (1 - 1)

= 0

Therefore, the integral of sin 2x from 0 lớn pi is 0.

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## What is the Integral of Sin^2x dx?

The integral of sin2x is denoted by ∫ sin2x dx & its value is (x/2) - (sin 2x)/4 + C. We can prove this in the following two methods.

By using the cos 2x formulaBy using the integration by parts

### Method 1: Integral of Sin^2x Using Double Angle Formula of Cos

To find the integral of sin2x, we use the double angle formula of cos. One of the cos 2x formulas is cos 2x = 1 - 2 sin2x. By solving this for sin2x, we get sin2x = (1 - cos 2x) / 2. We use this to lớn find ∫ sin2x dx. Then we get

∫ sin2x dx = ∫ (1 - cos 2x) / 2 dx

= (1/2) ∫ (1 - cos 2x) dx

= (1/2) ∫ 1 dx - (1/2) ∫ cos 2x dx

We know that ∫ cos 2x dx = (sin 2x)/2 + C. So

∫ sin2x dx = (1/2) x - (1/2) (sin 2x)/2 + C (or)

∫ sin2x dx = x/2 - (sin 2x)/4 + C

This is the integral of sin^2 x formula. Let us prove the same formula in another method.

### Method 2: Integral of Sin^2x Using Integration by Parts

We know that we can write sin2x as sin x · sin x. Khổng lồ find the integral of a product, we can use the integration by parts.

∫ sin2x dx = ∫ sin x · sin x dx = ∫ u dv

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

Then du = cos x dx và v = -cos x.

By integration by parts formula,

∫ u dv = uv - ∫ v du

∫ sin x · sin x dx = (sin x) (-cos x) - ∫ (-cos x)(cos x) dx

∫ sin2x dx = (-1/2) (2 sin x cos x) + ∫ cos2x dx

By the double angle formula of sin, 2 sin x cos x = sin 2x and by a trigonometric identity, cos2x = 1 - sin2x. So

∫ sin2x dx = (-1/2) sin 2x + ∫ (1 - sin2x) dx

∫ sin2x dx = (-1/2) sin 2x + ∫ 1 dx - ∫ sin2x dx

∫ sin2x dx + ∫ sin2x dx = (-1/2) sin 2x + x + C₁

2 ∫ sin2x dx = x - (1/2) sin 2x + C₁

∫ sin2x dx = x/2 - (sin 2x)/4 + C (where C = C₁/2)

Hence proved.

## Definite Integral of Sin^2x

To evaluate the definite integral of sin2x, we just substitute the upper and lower bounds in the value of the integral of sin2x & subtract the resultant values. Let us evaluate some definite integrals of integral sin2x dx here.

### Integral of Sin^2x From 0 to lớn 2pi

∫(_0^2pi) sin2x dx = (left. ight|_0^2pi)

= <2π/2 - (sin 4π)/4> - <0 - (sin 0)/4>

= π - 0/4

= π

Therefore, the integral of sin2x from 0 to 2π is π.

### Integral of Sin^2x From 0 lớn pi

∫(_0^pi) sin2x dx = (left. ight|_0^pi)

= <π/2 - (sin 2π)/4> - <0 - (sin 0)/4>

= π/2 - 0/4

= π/2

Therefore, the integral of sin2x from 0 lớn π is π/2.

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Important Notes Related khổng lồ Integral of Sin 2x & Integral of Sin2x:

∫ sin 2x dx = -(cos 2x)/2 + C∫ sin2x dx = x/2 - (sin 2x)/4 + C

Topics Related lớn Integral of Sin2x & Integral of Sin 2x:

Example 2: Evaluate the integral ∫ sin2x cos2x dx.

Solution:

By double angle formula of sin, 2 sin A cos A = sin 2A. Using this,

∫ sin2x cos2x dx = (1/4) ∫ (2 sin x cos x)2 dx

= (1/4) ∫ sin2(2x) dx

We have sin2x = (1 - cos 2x)/2. From this, sin22x = (1 - cos 4x)/2. So the above integral becomes

= (1/4) ∫ (1 - cos 4x)/2 dx

= (1/8) ∫ 1 dx - (1/8) ∫ cos 4x dx

To solve the second integral, we assume 4x = u, from which 4dx = du (or) dx = du/4. Then we get