Trigonometric Integrals Powers of Sine and Cosine cos^4(3x)
Trigonometric Integrals Powers of Sine and Cosine cos^4(3x)

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Integral of Cos4x

The integral of cos4x is equal to (1/4) sin4x + C, where C is the constant of integration. Let us recall the meaning of the integral of a function. The process of finding the anti-derivative of a function is called integration as it is the reverse process of differentiation. The integration of cos4x is the process of evaluating the antiderivative of cos4x (also called the integral of cos4x). The integral of a function gives the area under the curve of the function.

Let us evaluate the integral of cos4x using the substitution method, derive its formula, and find its definite integral as well with various limits using the formula. In this article, we will also determine the integration of cos^4x using the cos2x formula and algebraic identities. We shall solve various examples involving the integration of cos4x for a better understanding of the calculations involved.

1. What is the Integral of Cos4x?
2. Integration of Cos4x Formula
3. Integral of Cos4x Proof
4. Integration of Cos^4x
5. Definite Integral of Cos4x
6. FAQs on Integral of Cos4x

What is the Integral of Cos4x?

The integration of cos4x is equal to one-fourth of the trigonometric function sin 4x plus the constant of integration C. Mathematically, we can write the integral of cos4x as, ∫cos4x dx = (1/4) sin4x + C, where C is the integration constant. This integral can be evaluated using the substitution method of integration and the formula for the integral of cos x. The integration of cos4x gives the area under the curve of the function f(x) = cos4x. Cos4x is a trigonometric function of cosine with an angle four times x. Let us now explore the formula of cos4x in the next section.

Integration of Cos4x Formula

The formula for the integral of cos4x is given by, ∫cos4x dx = (1/4) sin4x + C. Here

  • ∫ is the symbol of integration
  • dx shows the integration is with respect to x, and
  • C is the constant of integration.

Using the integral of the cos4x formula, we can calculate the definite integration of cos4x by substituting the limits into the formula. The image given below shows the formula for the integral of cos4x:

Integral of Cos4x Proof

The integral of cos4x can be evaluated using the substitution method of integration. We know that the integral of cos x is equal to sin x + C. So, using this formula, we can derive the integration of cos4x. Assume 4x = u (an arbitrary variable). Now, differentiating both sides, we have 4dx = du. This implies dx = du/4. So, we have

∫cos 4x dx = ∫cos u du/4

= (1/4) ∫cos u du

= (1/4) [sin u + K] — [Integral of cos u is sin u + K, where K is the constant of integration]

= (1/4) sin u + K/4

= (1/4) sin 4x + C, where K/4 = C — [because u = 4x]

So, we have derived the formula for the integral of cos4x using the method of substitution in calculus.

Integration of Cos^4x

In this section, we will calculate the integral of cos^4x (that is, cos4x) using the cos2x formula in terms of the cosine function only. We know that we can cos4x as cos4x = (cos2x)2. The formula for cos2x is given by, cos2x = 2cos2x – 1 which implies cos2x = (cos2x + 1)/2. Therefore, we have

∫cos4x dx = ∫(cos2x)2 dx

= ∫ [(cos2x + 1)/2]2 dx

= ∫ (cos2x + 1)2/4 dx

= (1/4) ∫(cos22x + 1 + 2 cos2x) dx — [Using algebraic identity (a + b)2 = a2 + b2 + 2ab]

= (1/4) [ ∫ cos22x dx + ∫1 dx + ∫ 2 cos2x dx]

= (1/4) [ ∫(cos4x + 1)/2 dx + x + 2 ∫cos2x dx]

= (1/4) [ (1/2) ∫ (cos4x + 1) dx + x + 2 (sin2x) / 2]

= (1/4) [(1/2) (sin4x) / 4 + (1/2) x + x + sin 2x] + C

= (1/4) [(sin4x) / 8 + 3x/2 + sin2x] + C

= (1/32) sin4x + 3x/8 + (1/4) sin2x + C

Hence, we have determined the formula for the integral of cos^4x to be ∫cos4x dx = (1/32) sin4x + 3x/8 + (1/4) sin2x + C, where C is the integration constant.

Definite Integral of Cos4x

The definite integral of cos4x can be calculated by substituting the values of the limits into the formula of the integration of cos 4x. To find the definite integral a ∫b cos4x dx, we can derive its formula as,

a ∫b cos4x dx = [(1/4) sin 4x + C ]ab

= [(1/4) sin 4b + C] – [(1/4) sin 4a + C]

= (1/4) sin 4b + C – (1/4) sin 4a – C

= (1/4) (sin 4b – sin 4a)

Hence, the formula for the definite integral of cos4x is (1/4) (sin 4b – sin 4a)

Definite Integral of cos4x From 0 to Pi

To find the definite integral of cos4x with limits from 0 to pi, we will substitute these values into the formula. The formula for the definite integral of cos4x is (1/4) (sin 4b – sin 4a). Here b = π and a = 0. So, we have

0 ∫π cos4x dx = (1/4) (sin 4π – sin 4×0)

= (1/4) (0 – 0)

= 0

Therefore, the definite integral of cos 4x from 0 to pi is equal to 0.

Important Notes on Integral of cos4x

  • The integral of cos4x is given by, ∫cos4x dx = (1/4) sin4x + C where C is the constant of integration.
  • We can evaluate the integration of cos4x using the method of substitution in integration.
  • The formula for the definite integral of cos4x is a ∫b cos4x dx = (1/4) (sin 4b – sin 4a).

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Integral of Cos4x Examples

  1. Example 1: Evaluate the integral of cos4x cos3x.

    Solution: To find the integral of cos4x cos3x, we will use the formula cos A cos B = (1/2) [cos (A + B) + cos (A – B)]. Here A = 4x and B = 3x. So, we have cos4x cos3x = (1/2) [cos (4x + 3x) + cos (4x – 3x)] = (1/2) [cos 7x + cos x]. So, the required integral is given by,

    ∫ cos4x cos3x dx = ∫(1/2) [cos 7x + cos x] dx

    = (1/2) ∫cos7x dx + (1/2) ∫cos x dx

    = (1/2) (sin7x)/7 + (1/2) sin x + C —- [Because the integral of cos ax is equal to (sin ax) / a + K]

    = (1/14) sin7x + (1/2) sin x + C

    Answer: ∫ cos4x cos3x dx = (1/14) sin7x + (1/2) sin x + C

  2. Example 2: Calculate the definite integration of cos4x from 0 to pi/2.

    Solution: We know that the formula for the definite integration of cos4x is given by, a ∫b cos4x dx = (1/4) (sin 4b – sin 4a). Now, substitute a = 0 and b = π/2 into the formula.

    0 ∫π/2 cos4x dx = (1/4) (sin 4π/2 – sin 4(0))

    = (1/4) (sin 2π – sin 0)

    = (1/4) (0 – 0)

    = 0

    Answer: 0 ∫π/2 cos4x dx = 0

  3. Example 3: Find the integral of cos4x / sin2x.

    Solution: To find the integral of cos4x / sin2x, we will use the following formulas:

    • cos2A = 1 – 2sin2A
    • sin2x = 1/csc2x
    • Integral of sin2x: ∫sin2x dx = (-cos2x) / 2 + C
    • Intergal of cosec x: ∫ cosec x dx = ln |cosec x – cot x| + C

    So, we have

    ∫cos 4x / sin 2x dx = ∫ (1 – 2sin22x) / sin2x dx

    = ∫ (1/sin2x) dx – 2 ∫sin2x dx

    = ∫ csc2x dx – 2 (-cos2x) / 2 + C

    = (1/2) ln |cosec 2x – cot 2x| + cos2x + C

    Answer: ∫cos 4x / sin 2x dx = (1/2) ln |cosec 2x – cot 2x| + cos2x + C

Integration of Cos4x Questions

FAQs on Integral of Cos4x

What is Integral of Cos4x in Calculus?

The integral of cos4x is equal to (1/4) sin4x + C, where C is the constant of integration. The integration of cos4x gives the area under the curve of the function f(x) = cos4x.

How to Integrate Cos 4x?

The integral of cos4x can be evaluated using the substitution method of integration and the formula for the integral of cos x which is equal to sin x + C.

What is the Formula of Integration of Cos4x?

The formula for the integral of cos4x is given by, ∫cos4x dx = (1/4) sin4x + C, where C is the integration constant.

What is the Definite Integral of Cos4x?

The formula for the definite integral of cos4x is a ∫b cos4x dx = (1/4) (sin 4b – sin 4a).

What is the Integration of Cos 4x Cos x?

The integration of cos4x cos x is equal to (1/10) sin5x + (1/6) sin3x + C, where C is the constant of integration.

What is the Integration of Cos^4x?

The integral of cos^4x is given by, ∫cos4x dx = (1/32) sin4x + 3x/8 + (1/4) sin2x + C, where C is the integration constant.

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