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Identical expressions
sin^5x/cos^3xdx
sinus of to the power of 5x divide by co sinus of e of cubed xdx
sin5x/cos3xdx
sin⁵x/cos³xdx
sin to the power of 5x/cos to the power of 3xdx
sin^5x divide by cos^3xdx

Solving integrals/ sin^5x/cos^3xdx

Integral of sin^5x/cos^3xdx dx

The solution

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  1           
  /           
 |            
 |     5      
 |  sin (x)   
 |  ------- dx
 |     3      
 |  cos (x)   
 |            
/             
0

$$\int\limits_{0}^{1} \frac{\sin^{5}{\left(x \right)}}{\cos^{3}{\left(x \right)}}\, dx$$

Integral(sin(x)^5/cos(x)^3, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{\sin^{5}{\left(x \right)}}{\cos^{3}{\left(x \right)}} = \frac{\left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \cos^{2}{\left(x \right)}$ .
  
  Then let $du = - 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $- \frac{du}{2}$ :
  
  $\int \left(- \frac{u^{2} - 2 u + 1}{2 u^{2}}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u^{2} - 2 u + 1}{u^{2}}\, du = - \frac{\int \frac{u^{2} - 2 u + 1}{u^{2}}\, du}{2}$
    1. Rewrite the integrand:
      
      $\frac{u^{2} - 2 u + 1}{u^{2}} = 1 - \frac{2}{u} + \frac{1}{u^{2}}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{2}{u}\right)\, du = - 2 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- 2 \log{\left(u \right)}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      The result is: $u - 2 \log{\left(u \right)} - \frac{1}{u}$
    So, the result is: $- \frac{u}{2} + \log{\left(u \right)} + \frac{1}{2 u}$
  Now substitute $u$ back in:
  
  $\log{\left(\cos^{2}{\left(x \right)} \right)} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2 \cos^{2}{\left(x \right)}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} = \frac{\sin{\left(x \right)} \cos^{4}{\left(x \right)} - 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$
2. Let $u = \cos{\left(x \right)}$ .
  
  Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
  
  $\int \left(- \frac{u^{4} - 2 u^{2} + 1}{u^{3}}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u^{4} - 2 u^{2} + 1}{u^{3}}\, du = - \int \frac{u^{4} - 2 u^{2} + 1}{u^{3}}\, du$
    1. Let $u = u^{2}$ .
      
      Then let $du = 2 u du$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{u^{2} - 2 u + 1}{2 u^{2}}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u^{2} - 2 u + 1}{u^{2}}\, du = \frac{\int \frac{u^{2} - 2 u + 1}{u^{2}}\, du}{2}$
      
      Rewrite the integrand:
      
      $\frac{u^{2} - 2 u + 1}{u^{2}} = 1 - \frac{2}{u} + \frac{1}{u^{2}}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{2}{u}\right)\, du = - 2 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- 2 \log{\left(u \right)}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      The result is: $u - 2 \log{\left(u \right)} - \frac{1}{u}$
      
      So, the result is: $\frac{u}{2} - \log{\left(u \right)} - \frac{1}{2 u}$
      
      Now substitute $u$ back in:
      
      $\frac{u^{2}}{2} - \log{\left(u^{2} \right)} - \frac{1}{2 u^{2}}$
    So, the result is: $- \frac{u^{2}}{2} + \log{\left(u^{2} \right)} + \frac{1}{2 u^{2}}$
  Now substitute $u$ back in:
  
  $\log{\left(\cos^{2}{\left(x \right)} \right)} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2 \cos^{2}{\left(x \right)}}$
Method #3
1. Rewrite the integrand:
  
  $\frac{\left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} = \sin{\left(x \right)} \cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}$
2. Integrate term-by-term:
  1. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int \left(- u\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{2}{\left(x \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)\, dx = - 2 \int \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}\, dx$
    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: $2 \log{\left(\cos{\left(x \right)} \right)}$
  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: $2 \log{\left(\cos{\left(x \right)} \right)} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2 \cos^{2}{\left(x \right)}}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                   
 |                                                    
 |    5                            2                  
 | sin (x)              1       cos (x)      /   2   \
 | ------- dx = C + --------- - ------- + log\cos (x)/
 |    3                  2         2                  
 | cos (x)          2*cos (x)                         
 |                                                    
/

$$\int \frac{\sin^{5}{\left(x \right)}}{\cos^{3}{\left(x \right)}}\, dx = C + \log{\left(\cos^{2}{\left(x \right)} \right)} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2 \cos^{2}{\left(x \right)}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

0.335543178772137

0.335543178772137

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of sin^5x/cos^3xdx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3