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Integral of 1-sinx/sin^2x dx

The solution

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  1                 
  /                 
 |                  
 |  /     sin(x)\   
 |  |1 - -------| dx
 |  |       2   |   
 |  \    sin (x)/   
 |                  
/                   
0

$$\int\limits_{0}^{1} \left(- \frac{\sin{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 1\right)\, dx$$

Integral(1 - sin(x)/(sin(x)^2), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\sin{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)\, dx = - \int \frac{\sin{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2}$
  So, the result is: $- \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2}$
The result is: $x - \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$${{\log \left(\cos x+1\right)}\over{2}}-{{\log \left(\cos x-1\right) }\over{2}}+x$$

Simplify

The answer [src]

      pi*I
-oo - ----
       2

$${\it \%a}$$

      pi*I
-oo - ----
       2

$$-\infty - \frac{i \pi}{2}$$

Simplify

Numerical answer [src]

-43.1790108686112

-43.1790108686112

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

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Integral of 1-sinx/sin^2x dx

Limits of integration:

The graph:

Piecewise:

The solution