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Solving integrals/ sinxdx/(1-cosx)^3

Integral of sinxdx/(1-cosx)^3 dx

The solution

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 pi                          
  /                          
 |                           
 |                 1         
 |  sin(x)*1*------------- dx
 |                       3   
 |           (1 - cos(x))    
 |                           
/                            
pi                           
--                           
2

$$\int\limits_{\frac{\pi}{2}}^{\pi} \sin{\left(x \right)} 1 \cdot \frac{1}{\left(- \cos{\left(x \right)} + 1\right)^{3}}\, dx$$

Integral(sin(x)*1/(1 - cos(x))^3, (x, pi/2, pi))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\sin{\left(x \right)} 1 \cdot \frac{1}{\left(1 - \cos{\left(x \right)}\right)^{3}} = - \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)} - 3 \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 1}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)} - 3 \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 1}\right)\, dx = - \int \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)} - 3 \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 1}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{1}{2 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)} + 2}$
  So, the result is: $- \frac{1}{2 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)} + 2}$
Method #2
1. Rewrite the integrand:
  
  $\sin{\left(x \right)} 1 \cdot \frac{1}{\left(1 - \cos{\left(x \right)}\right)^{3}} = \frac{\sin{\left(x \right)}}{- \cos^{3}{\left(x \right)} + 3 \cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} + 1}$
2. Rewrite the integrand:
  
  $\frac{\sin{\left(x \right)}}{- \cos^{3}{\left(x \right)} + 3 \cos^{2}{\left(x \right)} - 3 \cos{\left(x \right)} + 1} = - \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)} - 3 \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 1}$
3. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)} - 3 \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 1}\right)\, dx = - \int \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)} - 3 \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 1}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{1}{2 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)} + 2}$
  So, the result is: $- \frac{1}{2 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)} + 2}$
Now simplify:

$- \frac{1}{2 \left(\cos{\left(x \right)} - 1\right)^{2}}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                        
 |                                                         
 |                1                           1            
 | sin(x)*1*------------- dx = C - ------------------------
 |                      3                              2   
 |          (1 - cos(x))           2 - 4*cos(x) + 2*cos (x)
 |                                                         
/

$$-{{1}\over{2\,\left(1-\cos x\right)^2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

3/8

$$\frac{3}{8}$$

3/8

$$\frac{3}{8}$$

Simplify

Numerical answer [src]

0.375

0.375

The graph

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

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Identical expressions

Similar expressions

Integral of sinxdx/(1-cosx)^3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2