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Derivative of:
cos^4(7x)
Identical expressions
cos^ four (7x)
co sinus of e of to the power of 4(7x)
co sinus of e of to the power of four (7x)
cos4(7x)
cos47x
cos⁴(7x)
cos^47x
cos^4(7x)dx

Integral of cos^4(7x) dx

The solution

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  1             
  /             
 |              
 |     4        
 |  cos (7*x) dx
 |              
/               
0

$$\int\limits_{0}^{1} \cos^{4}{\left(7 x \right)}\, dx$$

Integral(cos(7*x)^4, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\cos^{4}{\left(7 x \right)} = \left(\frac{\cos{\left(14 x \right)}}{2} + \frac{1}{2}\right)^{2}$
There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(\frac{\cos{\left(14 x \right)}}{2} + \frac{1}{2}\right)^{2} = \frac{\cos^{2}{\left(14 x \right)}}{4} + \frac{\cos{\left(14 x \right)}}{2} + \frac{1}{4}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos^{2}{\left(14 x \right)}}{4}\, dx = \frac{\int \cos^{2}{\left(14 x \right)}\, dx}{4}$
    1. Rewrite the integrand:
      
      $\cos^{2}{\left(14 x \right)} = \frac{\cos{\left(28 x \right)}}{2} + \frac{1}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(28 x \right)}}{2}\, dx = \frac{\int \cos{\left(28 x \right)}\, dx}{2}$
      
      Let $u = 28 x$ .
      
      Then let $du = 28 dx$ and substitute $\frac{du}{28}$ :
      
      $\int \frac{\cos{\left(u \right)}}{28}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{28}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{28}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(28 x \right)}}{28}$
      
      So, the result is: $\frac{\sin{\left(28 x \right)}}{56}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \frac{\sin{\left(28 x \right)}}{56}$
    So, the result is: $\frac{x}{8} + \frac{\sin{\left(28 x \right)}}{224}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(14 x \right)}}{2}\, dx = \frac{\int \cos{\left(14 x \right)}\, dx}{2}$
    1. Let $u = 14 x$ .
      
      Then let $du = 14 dx$ and substitute $\frac{du}{14}$ :
      
      $\int \frac{\cos{\left(u \right)}}{14}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{14}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{14}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(14 x \right)}}{14}$
    So, the result is: $\frac{\sin{\left(14 x \right)}}{28}$
  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: $\frac{3 x}{8} + \frac{\sin{\left(14 x \right)}}{28} + \frac{\sin{\left(28 x \right)}}{224}$
Method #2
1. Rewrite the integrand:
  
  $\left(\frac{\cos{\left(14 x \right)}}{2} + \frac{1}{2}\right)^{2} = \frac{\cos^{2}{\left(14 x \right)}}{4} + \frac{\cos{\left(14 x \right)}}{2} + \frac{1}{4}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos^{2}{\left(14 x \right)}}{4}\, dx = \frac{\int \cos^{2}{\left(14 x \right)}\, dx}{4}$
    1. Rewrite the integrand:
      
      $\cos^{2}{\left(14 x \right)} = \frac{\cos{\left(28 x \right)}}{2} + \frac{1}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(28 x \right)}}{2}\, dx = \frac{\int \cos{\left(28 x \right)}\, dx}{2}$
      
      Let $u = 28 x$ .
      
      Then let $du = 28 dx$ and substitute $\frac{du}{28}$ :
      
      $\int \frac{\cos{\left(u \right)}}{28}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{28}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{28}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(28 x \right)}}{28}$
      
      So, the result is: $\frac{\sin{\left(28 x \right)}}{56}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \frac{\sin{\left(28 x \right)}}{56}$
    So, the result is: $\frac{x}{8} + \frac{\sin{\left(28 x \right)}}{224}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(14 x \right)}}{2}\, dx = \frac{\int \cos{\left(14 x \right)}\, dx}{2}$
    1. Let $u = 14 x$ .
      
      Then let $du = 14 dx$ and substitute $\frac{du}{14}$ :
      
      $\int \frac{\cos{\left(u \right)}}{14}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{14}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{14}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(14 x \right)}}{14}$
    So, the result is: $\frac{\sin{\left(14 x \right)}}{28}$
  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: $\frac{3 x}{8} + \frac{\sin{\left(14 x \right)}}{28} + \frac{\sin{\left(28 x \right)}}{224}$
Add the constant of integration:

$\frac{3 x}{8} + \frac{\sin{\left(14 x \right)}}{28} + \frac{\sin{\left(28 x \right)}}{224}+ \mathrm{constant}$

The answer is:

$\frac{3 x}{8} + \frac{\sin{\left(14 x \right)}}{28} + \frac{\sin{\left(28 x \right)}}{224}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                              
 |                                               
 |    4               sin(14*x)   sin(28*x)   3*x
 | cos (7*x) dx = C + --------- + --------- + ---
 |                        28         224       8 
/

$$\int \cos^{4}{\left(7 x \right)}\, dx = C + \frac{3 x}{8} + \frac{\sin{\left(14 x \right)}}{28} + \frac{\sin{\left(28 x \right)}}{224}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       3                            
3   cos (7)*sin(7)   3*cos(7)*sin(7)
- + -------------- + ---------------
8         28                56

$$\frac{\sin{\left(7 \right)} \cos^{3}{\left(7 \right)}}{28} + \frac{3 \sin{\left(7 \right)} \cos{\left(7 \right)}}{56} + \frac{3}{8}$$

       3                            
3   cos (7)*sin(7)   3*cos(7)*sin(7)
- + -------------- + ---------------
8         28                56

$$\frac{\sin{\left(7 \right)} \cos^{3}{\left(7 \right)}}{28} + \frac{3 \sin{\left(7 \right)} \cos{\left(7 \right)}}{56} + \frac{3}{8}$$

3/8 + cos(7)^3*sin(7)/28 + 3*cos(7)*sin(7)/56

Numerical answer [src]

0.41158823497262

0.41158823497262

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(7x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2