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1/cos^3xsin^3xdx

Integral of 1/cos^3xsin^3x dx

The solution

You have entered [src]

  1           
  /           
 |            
 |     3      
 |  sin (x)   
 |  ------- dx
 |     3      
 |  cos (x)   
 |            
/             
0

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

Integral(sin(x)^3/cos(x)^3, (x, 0, 1))

Detail solution

Rewrite the integrand:

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

$\frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{2} + \frac{1}{2 \cos^{2}{\left(x \right)}}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{2} + \frac{1}{2 \cos^{2}{\left(x \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                                          
 |    3                            /   2   \
 | sin (x)              1       log\cos (x)/
 | ------- dx = C + --------- + ------------
 |    3                  2           2      
 | cos (x)          2*cos (x)               
 |                                          
/

$$\int \frac{\sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}}\, dx = C + \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{2} + \frac{1}{2 \cos^{2}{\left(x \right)}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1       1                  
- - + --------- + log(cos(1))
  2        2                 
      2*cos (1)

$$\log{\left(\cos{\left(1 \right)} \right)} - \frac{1}{2} + \frac{1}{2 \cos^{2}{\left(1 \right)}}$$

  1       1                  
- - + --------- + log(cos(1))
  2        2                 
      2*cos (1)

$$\log{\left(\cos{\left(1 \right)} \right)} - \frac{1}{2} + \frac{1}{2 \cos^{2}{\left(1 \right)}}$$

-1/2 + 1/(2*cos(1)^2) + log(cos(1))

Numerical answer [src]

0.597132940021366

0.597132940021366

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

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How to use it?

Identical expressions

Integral of 1/cos^3xsin^3x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3