Integral of Cos 2x

The integral of cos 2x and integration of cos²x are different and have the following values:

• ∫ cos 2x dx = (1/2) sin (2x) + C. To prove this, we use the substitution method of integration.
• ∫ cos²x dx = x/2 + (sin 2x)/4 + C. This is proved by using the identity of cos 2x followed by substitution method.

Let us find the integral of cos 2x and the integral of cos²x and also we will solve some problems related to these integrals.

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

∫ cos 2x dx = ∫ cos u (du/2)

= (1/2) ∫ cos u du

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

= (1/2) sin u + C

Substituting u = 2x back here,

∫ cos 2x dx = (1/2) sin (2x) + C

This is the integration of cos 2x formula.

A definite integral is nothing but an integral with lower and upper bounds. By the fundamental theorem of Calculus, to compute a definite integral, we substitute the upper bound first, and then the lower bound in the integral and then subtract them in the same order. In this process, we can ignore the integration constant. Let us calculate some definite integrals of integral cos 2x dx here.

Integral of Cos 2x From 0 to 2pi

∫₀^2π cos 2x dx = (1/2) sin (2x) |₀^2π

= (1/2) sin 2(2π) - (1/2) sin 2(0)

= (1/2) sin 4π - sin 0

= (1/2) 0 - 0

= 0

Therefore, the integration of cos 2x from 0 to 2pi is 0.

Integral of Cos 2x From 0 to pi

∫₀^π cos 2x dx = (1/2) sin (2x) |₀^π

= (1/2) sin 2(π) - sin 2(0)

= (1/2) sin 2π - sin 0

= 0 - 0

= 0

Therefore, the integral of cos 2x from 0 to pi is 0.

The integral of cos square x is denoted by ∫ cos²x dx and its value is (x/2) + (sin 2x)/4 + C. We can prove this in the following two methods.

• By using the cos 2x formula
• By using the integration by parts

Method 1: Integration of Cos^2x Using Double Angle Formula

To find the integral of cos²x, we use the double angle formula of cos. One of the cos 2x formulas is cos 2x = 2 cos²x - 1. By adding 1 on both sides, we get 1 + cos 2x = 2 cos²x. By dividing by both sides by 2, we get cos²x = (1 + cos 2x) / 2. We use this to find ∫ cos²x dx. Then we get

∫ cos²x dx = ∫ (1 + cos 2x) / 2 dx

= (1/2) ∫ (1 + cos 2x) dx

= (1/2) ∫ 1 dx + (1/2) ∫ cos 2x dx

In the previous sections, we have see that ∫ cos 2x dx = (sin 2x)/2 + C. So

∫ cos²x dx = (1/2) x + (1/2) (sin 2x)/2 + C (or)

∫ cos²x dx = x/2 + (sin 2x)/4 + C

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

Method 2: Integration of Cos^2x Using Integration by Parts

We know that we can write cos²x as cos x · cos x. Since it is a product, we can use the integration by parts to find the ∫ cos x · cos x dx. Then we get

∫ cos²x dx = ∫ cos x · cos x dx = ∫ u dv

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

Then du = - sin x dx and v = sin x.

By integration by parts formula,

∫ u dv = uv - ∫ v du

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

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

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

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

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

∫ cos²x dx + ∫ cos²x dx = (1/2) sin 2x + x + C₁

2 ∫ cos²x dx = (1/2) sin 2x + x + C₁

∫ cos²x dx = (1/4) sin 2x + x/2 + C (where C = C₁/2)

Hence proved.

To compute the definite integral of cos²x, we just substitute the upper and lower bounds in the value of the integral and subtract the resultant values. Let us calculate some definite integrals of integral cos²x dx here.

Integral of Cos^2x From 0 to 2pi

∫₀^2π cos²x dx = [x/2 + (sin 2x)/4] |₀^2π

= [2π/2 + (sin 4π)/4] - [0 + (sin 0)/4]

= π + 0/4

= π

Therefore, the integral of cos²x from 0 to 2π is π.

Integral of Cos^2x From 0 to pi

∫₀^π cos²x dx = [x/2 + (sin 2x)/4] |₀^π

= [π/2 + (sin 2π)/4] - [0 + (sin 0)/4]

= π/2 + 0/4

= π/2

Therefore, the integral of cos²x from 0 to π is π/2.

Important Notes on Integral of Cos 2x:

• ∫ cos 2x dx = (sin 2x)/2 + C
• ∫ cos²x dx = x/2 + (sin 2x)/4 + C

☛ Related Topics:

• Integral Calculator
• Indefinite Integral Calculator
• Integral of Sec x
• Applications of Integrals

What is the Integration of Cos 2x dx?

The integration of cos 2x dx is written as ∫ cos 2x dx and ∫ cos 2x dx = (sin 2x)/2 + C, where C is the integration constant.

What is the Integral Value of Cos^2x dx?

The integral of cos^2 x dx is written as ∫ cos²x dx and ∫ cos²dx = x/2 + (sin 2x)/4 + C. Here, C is the integration constant.

How to Find the Integral of Cos 2x?

The integral of cos 2x is found using the substitution method. In this, we assume that 2x = u, then 2 dx = du from which dx = du/2. Then the integral becomes (1/2) ∫ cos u du = (1/2) sin u + C = (1/2) sin 2x + C. Thus, the integral of cos 2x is (1/2) sin 2x + C, where 'C' is the integration constant.

How to Find the Definite Integration of Cos 2x from 0 to Pi?

We know that the integral of cos 2x is, ∫ cos 2x dx = (sin 2x)/2. Substituting the limits 0 and π, we get (1/2) sin 2(2π) - (1/2) sin 2(0) = (1/2) 0 - 0 = 0.

What is the Integral of Cos^3x dx?

∫ cos³x dx = ∫ cos²x cos x dx = ∫ (1 - sin²x) cos x dx. Let us substitute sin x = u. Then cos x dx = du. Then the above integral becomes ∫ (1 - u²) du = u - u³/3 + C. Substituting u = sin x back here, ∫ cos³x dx = sin x - sin³x/3 + C.

What is the Difference Between the Integration of Cos 2x and the Integral of Cos x?

• The integral of cos 2x is ∫ cos 2x dx = (sin 2x)/2 + C.
• The integral of cos x is ∫ cos x dx = sin x + C.

What is the Integral of Cos 3x dx?

To find the ∫ cos 3x dx, we assume that 3x = u. Then 3 dx = du (or) dx = du/3. Then the above integral becomes, ∫ cos u (1/3) du = (1/3) sin u + C = (1/3) sin (3x) + C.

Is the Integral Cos 2x dx Same as Integral Cos Square x dx?

No, the values of these two integrals are NOT same. We have

• The integral of cos square x is, ∫ cos²dx = x/2 + (sin 2x)/4 + C
• The integral of cos 2x is, ∫ cos 2x dx = (sin 2x)/2 + C

How to Find the Definite Integral of Cos^2x from 0 to 2Pi?

We know that ∫ cos²dx = x/2 + (sin 2x)/4 + C. Substituting the limits here, we get the value of the definite integral to be [π/2 + (sin 2π)/4] - [0 + (sin 0)/4] = π/2.