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Integral of cos^2x
Integral of cos^2x

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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.

1. What is the Integral of Cos 2x dx?
2. Definite Integral of Cos 2x
3. What is the Integral of Cos^2x dx?
4. Definite Integral of Cos^2x
5. FAQs on Integral of Cos 2x and Cos^2x

What is the Integral of Cos 2x dx?

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.

Definite Integral of Cos 2x

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

∫02π cos 2x dx = (1/2) sin (2x) |02π

= (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

∫0π cos 2x dx = (1/2) sin (2x) |0π

= (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.

Integration of Sin^2x dx

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.

Definite Integral of Cos^2x

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

∫02π cos2x dx = [x/2 + (sin 2x)/4] |02π

= [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

∫0π cos2x dx = [x/2 + (sin 2x)/4] |0π

= [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:

Examples of Integration of Cos 2x and Cos^2x

  1. Example 1: Evaluate the integral ∫ cos 2x esin 2x dx.

    Solution:

    We will solve this using the substitution method.

    Let sin 2x = u. Then 2 cos 2x dx = du (or) cos 2x dx = (1/2) du.

    Then the given integral becomes,

    ∫ eu (1/2) du = (1/2) eu + C.

    Let us substitute the value of u = sin 2x back here. Then we get

    ∫ cos 2x esin 2x dx = (1/2) esin 2x + C.

    Answer: The integration of cos 2x e^sin 2x dx is (1/2) esin 2x + C.

  2. Example 2: Evaluate the integration ∫ cos 2x sin 2x dx.

    Solution:

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

    ∫ cos 2x sin 2x dx = (1/2) ∫ 2 sin 2x cos 2x dx

    = (1/2) ∫ sin 2(2x) dx

    = (1/2) ∫ sin 4x dx

    Let 4x = u. Then 4 dx = du (or) dx = du/4. Then

    ∫ cos 2x sin 2x dx = (1/2) ∫ sin u (du/4)

    = (1/8) ∫ sin u du

    = (1/8) (- cos u) + C

    = (1/8) (- cos 2x) + C

    Answer: The integral of integral cos 2x sin 2x dx is (-1/8) (cos 2x) + C.

  3. Example 3: Evaluate the integral ∫ (1 + cos x)2 dx.

    Solution:

    ∫ (1 + cos x)2 dx = ∫ (1 + 2 cos x + cos2x) dx

    = ∫ 1 dx + 2 ∫ cos x dx + ∫ cos2dx

    We know that ∫ cos x dx = sin x and ∫ cos2x dx = x/2 + (sin 2x)/4. So the above integral becomes

    = x + 2 sin x + x/2 + (sin 2x)/4

    = (3x)/2 + (1/4) sin 2x + 2 sin x + C.

    Answer: The integral of (1 + cos x)^2 is (3x)/2 + (1/4) sin 2x + 2 sin x + C.

Practice Questions on Integral of Cos 2x and Cos^2x

FAQs on Integral of Cos 2x and Cos^2x

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.

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