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((1+sinx)cosx)/sinx

Integral of ((1-sinx)cosx)/sinx dx

The solution

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

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

Integral(((1 - sin(x))*cos(x))/sin(x), (x, 0, 1))

Detail solution

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

$\log{\left(- \sin{\left(x \right)} \right)} - \sin{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(- \sin{\left(x \right)} \right)} - \sin{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

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$$\infty$$

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Numerical answer [src]

43.0763714029159

43.0763714029159

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

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Identical expressions

Similar expressions

Integral of ((1-sinx)cosx)/sinx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3