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Integral of d{x}:
Integral of sin(3x)*cos(5x)
Integral of 1/(4sin^2x+9cos^2x)
Integral of sint/(2-cost)
Integral of sqrt(x^2+4x-13)
Identical expressions
sint/(two -cost)
sinus of t divide by (2 minus co sinus of e of t)
sinus of t divide by (two minus co sinus of e of t)
sint/2-cost
sint divide by (2-cost)
Similar expressions
sint/(2+cost)

Integral of sint/(2-cost) dx

The solution

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          0                      
          /                      
         |                       
         |            sin(t)     
         |          ---------- dt
         |          2 - cos(t)   
         |                       
        /                        
POST_GRBEK_SMALL_pi

$$\int\limits_{POST_{GRBEK SMALL \pi}}^{0} \frac{\sin{\left(t \right)}}{2 - \cos{\left(t \right)}}\, dt$$

Integral(sin(t)/(2 - cos(t)), (t, POST_GRBEK_SMALL_pi, 0))

Detail solution

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

$\log{\left(2 - \cos{\left(t \right)} \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(2 - \cos{\left(t \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                    
 |   sin(t)                           
 | ---------- dt = C + log(2 - cos(t))
 | 2 - cos(t)                         
 |                                    
/

$$\int \frac{\sin{\left(t \right)}}{2 - \cos{\left(t \right)}}\, dt = C + \log{\left(2 - \cos{\left(t \right)} \right)}$$

Simplify

The answer [src]

-log(-2 + cos(POST_GRBEK_SMALL_pi)) + pi*I

$$- \log{\left(\cos{\left(POST_{GRBEK SMALL \pi} \right)} - 2 \right)} + i \pi$$

-log(-2 + cos(POST_GRBEK_SMALL_pi)) + pi*I

$$- \log{\left(\cos{\left(POST_{GRBEK SMALL \pi} \right)} - 2 \right)} + i \pi$$

Упростить

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

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How to use it?

Identical expressions

Similar expressions

Integral of sint/(2-cost) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2