Integral of cos^37x dx

The solution

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 |     3        
 |  cos (7*x) dx
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0

$$\int\limits_{0}^{1} \cos^{3}{\left(7 x \right)}\, dx$$

Integral(cos(7*x)^3, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\cos^{3}{\left(7 x \right)} = \left(1 - \sin^{2}{\left(7 x \right)}\right) \cos{\left(7 x \right)}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \sin{\left(7 x \right)}$ .
  
  Then let $du = 7 \cos{\left(7 x \right)} dx$ and substitute $du$ :
  
  $\int \left(\frac{1}{7} - \frac{u^{2}}{7}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{7}\, du = \frac{u}{7}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u^{2}}{7}\right)\, du = - \frac{\int u^{2}\, du}{7}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{21}$
    The result is: $- \frac{u^{3}}{21} + \frac{u}{7}$
  Now substitute $u$ back in:
  
  $- \frac{\sin^{3}{\left(7 x \right)}}{21} + \frac{\sin{\left(7 x \right)}}{7}$
Method #2
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(7 x \right)}\right) \cos{\left(7 x \right)} = - \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)} + \cos{\left(7 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}\right)\, dx = - \int \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}\, dx$
    1. Let $u = \sin{\left(7 x \right)}$ .
      
      Then let $du = 7 \cos{\left(7 x \right)} dx$ and substitute $\frac{du}{7}$ :
      
      $\int \frac{u^{2}}{49}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u^{2}}{7}\, du = \frac{\int u^{2}\, du}{7}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $\frac{u^{3}}{21}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(7 x \right)}}{21}$
    So, the result is: $- \frac{\sin^{3}{\left(7 x \right)}}{21}$
  1. Let $u = 7 x$ .
    
    Then let $du = 7 dx$ and substitute $\frac{du}{7}$ :
    
    $\int \frac{\cos{\left(u \right)}}{49}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{7}\, du = \frac{\int \cos{\left(u \right)}\, du}{7}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{7}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(7 x \right)}}{7}$
  The result is: $- \frac{\sin^{3}{\left(7 x \right)}}{21} + \frac{\sin{\left(7 x \right)}}{7}$
Method #3
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(7 x \right)}\right) \cos{\left(7 x \right)} = - \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)} + \cos{\left(7 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}\right)\, dx = - \int \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}\, dx$
    1. Let $u = \sin{\left(7 x \right)}$ .
      
      Then let $du = 7 \cos{\left(7 x \right)} dx$ and substitute $\frac{du}{7}$ :
      
      $\int \frac{u^{2}}{49}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u^{2}}{7}\, du = \frac{\int u^{2}\, du}{7}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $\frac{u^{3}}{21}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(7 x \right)}}{21}$
    So, the result is: $- \frac{\sin^{3}{\left(7 x \right)}}{21}$
  1. Let $u = 7 x$ .
    
    Then let $du = 7 dx$ and substitute $\frac{du}{7}$ :
    
    $\int \frac{\cos{\left(u \right)}}{49}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{7}\, du = \frac{\int \cos{\left(u \right)}\, du}{7}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{7}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(7 x \right)}}{7}$
  The result is: $- \frac{\sin^{3}{\left(7 x \right)}}{21} + \frac{\sin{\left(7 x \right)}}{7}$
Now simplify:

$\frac{3 \sin{\left(7 x \right)}}{28} + \frac{\sin{\left(21 x \right)}}{84}$
Add the constant of integration:

$\frac{3 \sin{\left(7 x \right)}}{28} + \frac{\sin{\left(21 x \right)}}{84}+ \mathrm{constant}$

The answer is:

$\frac{3 \sin{\left(7 x \right)}}{28} + \frac{\sin{\left(21 x \right)}}{84}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                       3                
 |    3               sin (7*x)   sin(7*x)
 | cos (7*x) dx = C - --------- + --------
 |                        21         7    
/

$${{\sin \left(7\,x\right)-{{\sin ^3\left(7\,x\right)}\over{3}} }\over{7}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

     3            
  sin (7)   sin(7)
- ------- + ------
     21       7

$$-{{\sin ^37-3\,\sin 7}\over{21}}$$

     3            
  sin (7)   sin(7)
- ------- + ------
     21       7

$$- \frac{\sin^{3}{\left(7 \right)}}{21} + \frac{\sin{\left(7 \right)}}{7}$$

Упростить

Numerical answer [src]

0.0803516074643471

0.0803516074643471

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of cos^37x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3