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Similar expressions
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Integral of sinx-xcosx/sin^2x dx

The solution

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

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

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

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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$$-\infty$$

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$$-\infty$$

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Numerical answer [src]

-43.5309180687013

-43.5309180687013

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

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Identical expressions

Similar expressions

Integral of sinx-xcosx/sin^2x dx

Limits of integration:

The graph:

Piecewise:

The solution