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Integral of d{x}:
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Integral of x-2
Identical expressions
cos^ four (x/ eight)
co sinus of e of to the power of 4(x divide by 8)
co sinus of e of to the power of four (x divide by eight)
cos4(x/8)
cos4x/8
cos⁴(x/8)
cos^4x/8
cos^4(x divide by 8)
cos^4(x/8)dx

Integral of cos^4(x/8) dx

The solution

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$$\int\limits_{0}^{0} \cos^{4}{\left(\frac{x}{8} \right)}\, dx$$

Integral(cos(x/8)^4, (x, 0, 0))

Detail solution

Rewrite the integrand:

$\cos^{4}{\left(\frac{x}{8} \right)} = \left(\frac{\cos{\left(\frac{x}{4} \right)}}{2} + \frac{1}{2}\right)^{2}$
There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(\frac{\cos{\left(\frac{x}{4} \right)}}{2} + \frac{1}{2}\right)^{2} = \frac{\cos^{2}{\left(\frac{x}{4} \right)}}{4} + \frac{\cos{\left(\frac{x}{4} \right)}}{2} + \frac{1}{4}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos^{2}{\left(\frac{x}{4} \right)}}{4}\, dx = \frac{\int \cos^{2}{\left(\frac{x}{4} \right)}\, dx}{4}$
    1. Rewrite the integrand:
      
      $\cos^{2}{\left(\frac{x}{4} \right)} = \frac{\cos{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(\frac{x}{2} \right)}}{2}\, dx = \frac{\int \cos{\left(\frac{x}{2} \right)}\, dx}{2}$
      
      Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 2 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $2 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $2 \sin{\left(\frac{x}{2} \right)}$
      
      So, the result is: $\sin{\left(\frac{x}{2} \right)}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \sin{\left(\frac{x}{2} \right)}$
    So, the result is: $\frac{x}{8} + \frac{\sin{\left(\frac{x}{2} \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(\frac{x}{4} \right)}}{2}\, dx = \frac{\int \cos{\left(\frac{x}{4} \right)}\, dx}{2}$
    1. Let $u = \frac{x}{4}$ .
      
      Then let $du = \frac{dx}{4}$ and substitute $4 du$ :
      
      $\int 4 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 4 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $4 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $4 \sin{\left(\frac{x}{4} \right)}$
    So, the result is: $2 \sin{\left(\frac{x}{4} \right)}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{4}\, dx = \frac{x}{4}$
  The result is: $\frac{3 x}{8} + 2 \sin{\left(\frac{x}{4} \right)} + \frac{\sin{\left(\frac{x}{2} \right)}}{4}$
Method #2
1. Rewrite the integrand:
  
  $\left(\frac{\cos{\left(\frac{x}{4} \right)}}{2} + \frac{1}{2}\right)^{2} = \frac{\cos^{2}{\left(\frac{x}{4} \right)}}{4} + \frac{\cos{\left(\frac{x}{4} \right)}}{2} + \frac{1}{4}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos^{2}{\left(\frac{x}{4} \right)}}{4}\, dx = \frac{\int \cos^{2}{\left(\frac{x}{4} \right)}\, dx}{4}$
    1. Rewrite the integrand:
      
      $\cos^{2}{\left(\frac{x}{4} \right)} = \frac{\cos{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(\frac{x}{2} \right)}}{2}\, dx = \frac{\int \cos{\left(\frac{x}{2} \right)}\, dx}{2}$
      
      Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 2 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $2 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $2 \sin{\left(\frac{x}{2} \right)}$
      
      So, the result is: $\sin{\left(\frac{x}{2} \right)}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \sin{\left(\frac{x}{2} \right)}$
    So, the result is: $\frac{x}{8} + \frac{\sin{\left(\frac{x}{2} \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(\frac{x}{4} \right)}}{2}\, dx = \frac{\int \cos{\left(\frac{x}{4} \right)}\, dx}{2}$
    1. Let $u = \frac{x}{4}$ .
      
      Then let $du = \frac{dx}{4}$ and substitute $4 du$ :
      
      $\int 4 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 4 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $4 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $4 \sin{\left(\frac{x}{4} \right)}$
    So, the result is: $2 \sin{\left(\frac{x}{4} \right)}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{4}\, dx = \frac{x}{4}$
  The result is: $\frac{3 x}{8} + 2 \sin{\left(\frac{x}{4} \right)} + \frac{\sin{\left(\frac{x}{2} \right)}}{4}$
Add the constant of integration:

$\frac{3 x}{8} + 2 \sin{\left(\frac{x}{4} \right)} + \frac{\sin{\left(\frac{x}{2} \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{3 x}{8} + 2 \sin{\left(\frac{x}{4} \right)} + \frac{\sin{\left(\frac{x}{2} \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               /x\      
 |                             sin|-|      
 |    4/x\               /x\      \2/   3*x
 | cos |-| dx = C + 2*sin|-| + ------ + ---
 |     \8/               \4/     4       8 
 |                                         
/

$$\int \cos^{4}{\left(\frac{x}{8} \right)}\, dx = C + \frac{3 x}{8} + 2 \sin{\left(\frac{x}{4} \right)} + \frac{\sin{\left(\frac{x}{2} \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

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Use the examples entering the upper and lower limits of integration.

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How to use it?

Identical expressions

Integral of cos^4(x/8) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2