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Solving integrals/ (x+cosx)/(x+2sinx)

Integral of (x+cosx)/(x+2sinx) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src] 
 2*pi               
   /                
  |                 
  |   x + cos(x)    
  |  ------------ dx
  |  x + 2*sin(x)   
  |                 
 /                  
 pi
∫π2πx+cos⁡(x)x+2sin⁡(x) dx\int\limits_{\pi}^{2 \pi} \frac{x + \cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx 
Integral((x + cos(x))/(x + 2*sin(x)), (x, pi, 2*pi))
Detail solution

  1. Rewrite the integrand:

    x+cos⁡(x)x+2sin⁡(x)=xx+2sin⁡(x)+cos⁡(x)x+2sin⁡(x)\frac{x + \cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}} = \frac{x}{x + 2 \sin{\left(x \right)}} + \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}

  2. Integrate term-by-term:

    1. Don't know the steps in finding this integral.

      But the integral is

      ∫xx+2sin⁡(x) dx\int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx

    1. Don't know the steps in finding this integral.

      But the integral is

      ∫cos⁡(x)x+2sin⁡(x) dx\int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx

    The result is: ∫xx+2sin⁡(x) dx+∫cos⁡(x)x+2sin⁡(x) dx\int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx + \int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx

  3. Add the constant of integration:

    ∫xx+2sin⁡(x) dx+∫cos⁡(x)x+2sin⁡(x) dx+constant\int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx + \int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx+ \mathrm{constant}

The answer is:

∫xx+2sin⁡(x) dx+∫cos⁡(x)x+2sin⁡(x) dx+constant\int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx + \int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                        /                    /               
 |                        |                    |                
 |  x + cos(x)            |      x             |    cos(x)      
 | ------------ dx = C +  | ------------ dx +  | ------------ dx
 | x + 2*sin(x)           | x + 2*sin(x)       | x + 2*sin(x)   
 |                        |                    |                
/                        /                    /
∫x+cos⁡(x)x+2sin⁡(x) dx=C+∫xx+2sin⁡(x) dx+∫cos⁡(x)x+2sin⁡(x) dx\int \frac{x + \cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx = C + \int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx + \int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx
Simplify
The answer [src] 
 2*pi               
   /                
  |                 
  |   x + cos(x)    
  |  ------------ dx
  |  x + 2*sin(x)   
  |                 
 /                  
 pi
∫π2πx+cos⁡(x)x+2sin⁡(x) dx\int\limits_{\pi}^{2 \pi} \frac{x + \cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx 
=
= 
 2*pi               
   /                
  |                 
  |   x + cos(x)    
  |  ------------ dx
  |  x + 2*sin(x)   
  |                 
 /                  
 pi
∫π2πx+cos⁡(x)x+2sin⁡(x) dx\int\limits_{\pi}^{2 \pi} \frac{x + \cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx 
Integral((x + cos(x))/(x + 2*sin(x)), (x, pi, 2*pi))
Numerical answer [src] 
4.33494741276426
4.33494741276426

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

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